Volume 249

Graduate Texts in Mathematics

Series EditorsSheldon Axler and Kenneth Ribet

Graduate Texts in Mathematics bridge the gap between passive study and creative understanding, offering graduate-level introductions to advanced topics in mathematics. The volumes are carefully written as teaching aids and highlight characteristic features of the theory. Although these books are frequently used as textbooks in graduate courses, they are also suitable for individual study.

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Loukas Grafakos

Classical Fourier Analysis

Loukas Grafakos

Department of Mathematics, University of Missouri, Columbia, MO, USA

ISSN 0072-5285e-ISSN 2197-5612

ISBN 978-1-4939-1193-6e-ISBN 978-1-4939-1194-3

DOI 10.1007/978-1-4939-1194-3

Springer New York Heidelberg Dordrecht London

Library of Congress Control Number: 2014946585

Mathematics Subject Classiﬁcation (2010): 42Axx42Bxx

© Springer Science+Business Media New York 2014

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To Suzanne

Preface

The great response to the publication of my book Classical and Modern Fourier Analysis in 2004 has been especially gratifying to me. I was delighted when Springer offered to publish the second edition in 2008 in two volumes: Classical Fourier Analysis , 2 nd Edition , and Modern Fourier Analysis , 2 nd Edition . I am now elated to have the opportunity to write the present third edition of these books, which Springer has also kindly offered to publish. The third edition was born from my desire to improve the exposition in several places, fix a few inaccuracies, and add some new material. I have been very fortunate to receive several hundred e-mail messages that helped me improve the proofs and locate mistakes and misprints in the previous editions.

In this edition, I maintain the same style as in the previous ones. The proofs contain details that unavoidably make the reading more cumbersome. Although it will behoove many readers to skim through the more technical aspects of the presentation and concentrate on the flow of ideas, the fact that details are present will be comforting to some. (This last sentence is based on my experience as a graduate student.) Readers familiar with the second edition will notice that the chapter on weights has been moved from the second volume to the first.

This first volume Classical Fourier Analysis is intended to serve as a text for a one-semester course with prerequisites of measure theory, Lebesgue integration, and complex variables. I am aware that this book contains significantly more material than can be taught in a semester course; however, I hope that this additional information will be useful to researchers. Based on my experience, the following list of sections (or parts of them) could be taught in a semester without affecting the logical coherence of the book: Sections 1.​1 , 1.​2 , 1.​3 , 2.​1 , 2.​2 ., 2.​3 , 3.​1 , 3.​2 , 3.​3 , 4.​4 , 4.​5 , 5.​1 , 5.​2 , 5.​3 , 5.​5 , 5.​6 , 6.​1 , 6.​2 .

A long list of people have assisted me in the preparation of this book, but I remain solely responsible for any misprints, mistakes, and omissions contained therein. Please contact me directly (grafakosl@missouri.edu) if you have corrections or comments. Any corrections to this edition will be posted to the website

http://math.missouri.edu/~loukas/FourierAnalysis.html

which I plan to update regularly. I have prepared solutions to all of the exercises for the present edition which will be available to instructors who teach a course out of this book.

Loukas Grafakos

Athens, Greece

March 2014

Acknowledgments

I am extremely fortunate that several people have pointed out errors, misprints, and omissions in the previous editions of the books in this series. All these individuals have provided me with invaluable help that resulted in the improved exposition of the text. For these reasons, I would like to express my deep appreciation and sincere gratitude to the all of the following people.

First edition acknowledgements: Georgios Alexopoulos, Nakhlé Asmar, Bruno Calado, Carmen Chicone, David Cramer, Geoffrey Diestel, Jakub Duda, Brenda Frazier, Derrick Hart, Mark Hoffmann, Steven Hofmann, Helge Holden, Brian Hollenbeck, Petr Honzík, Alexander Iosevich, Tunde Jakab, Svante Janson, Ana Jiménez del Toro, Gregory Jones, Nigel Kalton, Emmanouil Katsoprinakis, Dennis Kletzing, Steven Krantz, Douglas Kurtz, George Lobell, Xiaochun Li, José María Martell, Antonios Melas, Keith Mersman, Stephen Montgomery-Smith, Andrea Nahmod, Nguyen Cong Phuc, Krzysztof Oleszkiewicz, Cristina Pereyra, Carlos Pérez, Daniel Redmond, Jorge Rivera-Noriega, Dmitriy Ryabogin, Christopher Sansing, Lynn Savino Wendel, Shih-Chi Shen, Roman Shvidkoy, Elias M. Stein, Atanas Stefanov, Terence Tao, Erin Terwilleger, Christoph Thiele, Rodolfo Torres, Deanie Tourville, Nikolaos Tzirakis, Don Vaught, Igor Verbitsky, Brett Wick, James Wright, and Linqiao Zhao.

Second edition acknowledgements: Marco Annoni, Pascal Auscher, Andrew Bailey, Dmitriy Bilyk, Marcin Bownik, Juan Cavero de Carondelet Fiscowich, Leonardo Colzani, Simon Cowell, Mita Das, Geoffrey Diestel, Yong Ding, Jacek Dziubanski, Frank Ganz, Frank McGuckin, Wei He, Petr Honzík, Heidi Hulsizer, Philippe Jaming, Svante Janson, Ana Jiménez del Toro, John Kahl, Cornelia Kaiser, Nigel Kalton, Kim Jin Myong, Doowon Koh, Elena Koutcherik, David Kramer, Enrico Laeng, Sungyun Lee, Qifan Li, Chin-Cheng Lin, Liguang Liu, Stig-Olof Londen, Diego Maldonado, José María Martell, Mieczysław Mastyło, Parasar Mohanty, Carlo Morpurgo, Andrew Morris, Mihail Mourgoglou, Virginia Naibo, Tadahiro Oh, Marco Peloso, Maria Cristina Pereyra, Carlos Pérez, Humberto Rafeiro, Maria Carmen Reguera Rodríguez, Alexander Samborskiy, Andreas Seeger, Steven Senger, Sumi Seo, Christopher Shane, Shu Shen, Yoshihiro Sawano, Mark Spencer, Vladimir Stepanov, Erin Terwilleger, Rodolfo H. Torres, Suzanne Tourville, Ignacio Uriarte-Tuero, Kunyang Wang, Huoxiong Wu, Kôzô Yabuta, Takashi Yamamoto, and Dachun Yang.

Third edition acknowledgments: Marco Annoni, Mark Ashbaugh, Daniel Azagra, Andrew Bailey, Árpad Bényi, Dmitriy Bilyk, Nicholas Boros, Almut Burchard, María Carro, Jameson Cahill, Juan Cavero de Carondelet Fiscowich, Xuemei Chen, Andrea Fraser, Shai Dekel, Fausto Di Biase, Zeev Ditzian, Jianfeng Dong, Oliver Dragičević, Sivaji Ganesh, Friedrich Gesztesy, Zhenyu Guo, Piotr Hajłasz, Danqing He, Andreas Heinecke, Steven Hofmann, Takahisa Inui, Junxiong Jia, Kasinathan Kamalavasanthi, Hans Koelsch, Richard Laugesen, Kaitlin Leach, Andrei Lerner, Yiyu Liang, Calvin Lin, Liguang Liu, Elizabeth Loew, Chao Lu, Richard Lynch, Diego Maldonado, Lech Maligranda, Richard Marcum, Mieczysław Mastyło, Mariusz Mirek, Carlo Morpurgo, Virginia Naibo, Hanh Van Nguyen, Seungly Oh, Tadahiro Oh, Yusuke Oi, Lucas da Silva Oliveira, Kevin OʼNeil, Hesam Oveys, Manos Papadakis, Marco Peloso, Carlos Pérez, Jesse Peterson, Dmitry Prokhorov, Amina Ravi, Maria Carmen Reguera Rodríguez, Yoshihiro Sawano, Mirye Shin, Javier Soria, Patrick Spencer, Marc Strauss, Krystal Taylor, Naohito Tomita, Suzanne Tourville, Rodolfo H. Torres, Fujioka Tsubasa, Ignacio Uriarte-Tuero, Brian Tuomanen, Shibi Vasudevan, Michael Wilson, Dachun Yang, Kai Yang, Yandan Zhang, Fayou Zhao, and Lifeng Zhao.

Among all these people, I would like to give special thanks to an individual who has studied extensively the two books in the series and has helped me more than anyone else in the preparation of the third edition: Danqing He. I am indebted to him for all the valuable corrections, suggestions, and constructive help he has provided me with in this work. Without him, these books would have been a lot poorer.

Finally, I would also like to thank the University of Missouri for granting me a research leave during the academic year 2013-2014. This time off enabled me to finish the third edition of this book on time. I spent my leave in Greece.

Contents

1 L p Spaces and Interpolation 1

1.1 L p and Weak L p 1

1.​1.​1 The Distribution Function 3

1.​1.​2 Convergence in Measure 6

1.​1.​3 A First Glimpse at Interpolation 9

Exercises 11

1.​2 Convolution and Approximate Identities 17

1.​2.​1 Examples of Topological Groups 18

1.​2.​2 Convolution 19

1.​2.​3 Basic Convolution Inequalities 21

1.​2.​4 Approximate Identities 25

Exercises 30

1.​3 Interpolation 33

1.​3.​1 Real Method:​ The Marcinkiewicz Interpolation Theorem 33

1.​3.​2 Complex Method:​ The Riesz–Thorin Interpolation Theorem 36

1.​3.​3 Interpolation of Analytic Families of Operators 40

Exercises 45

1.​4 Lorentz Spaces 48

1.​4.​1 Decreasing Rearrangements 48

1.​4.​2 Lorentz Spaces 52

1.​4.​3 Duals of Lorentz Spaces 56

1.​4.​4 The Off-Diagonal Marcinkiewicz Interpolation Theorem 60

Exercises 74

2 Maximal Functions, Fourier Transform, and Distributions 85

2.​1 Maximal Functions 86

2.​1.​1 The Hardy–Littlewood Maximal Operator 86

2.​1.​2 Control of Other Maximal Operators 90

2.​1.​3 Applications to Differentiation Theory 93

Exercises 98

2.​2 The Schwartz Class and the Fourier Transform 104

2.​2.​1 The Class of Schwartz Functions 105

2.​2.​2 The Fourier Transform of a Schwartz Function 108

2.​2.​3 The Inverse Fourier Transform and Fourier Inversion 111

2.2.4 The Fourier Transform on $$L^{1} + L^{2}$$ 113

Exercises 116

2.​3 The Class of Tempered Distributions 119

2.​3.​1 Spaces of Test Functions 119

2.​3.​2 Spaces of Functionals on Test Functions 120

2.​3.​3 The Space of Tempered Distributions 123

Exercises 131

2.​4 More About Distributions and the Fourier Transform 133

2.​4.​1 Distributions Supported at a Point 134

2.​4.​2 The Laplacian 135

2.​4.​3 Homogeneous Distributions 136

Exercises 143

2.5 Convolution Operators on $$L^{p}$$ Spaces and Multipliers 146

2.​5.​1 Operators That Commute with Translations 146

2.​5.​2 The Transpose and the Adjoint of a Linear Operator 150

2.5.3 The Spaces $$\mathcal{M}^{p,q}(\mathbf{R}^{n})$$ 151

2.5.4 Characterizations of $$\mathcal{M}^{1,1}(\mathbf{R}^{n})$$ and $$\mathcal{M}^{2,2}(\mathbf{R}^{n})$$ 153

2.5.5 The Space of Fourier Multipliers $$\mathcal{M}:{p}(\mathbf{R}^{n})$$ 155

Exercises 159

2.​6 Oscillatory Integrals 161

2.​6.​1 Phases with No Critical Points 161

2.​6.​2 Sublevel Set Estimates and the Van der Corput Lemma 164

Exercises 169

3 Fourier Series 173

3.​1 Fourier Coefficients 173

3.1.1 The n -Torus $$\mathbf{T}^{n}$$ 174

3.​1.​2 Fourier Coefficients 175

3.​1.​3 The Dirichlet and Fejér Kernels 178

Exercises 182

3.​2 Reproduction of Functions from Their Fourier Coefficients 183

3.​2.​1 Partial sums and Fourier inversion 183

3.​2.​2 Fourier series of square summable functions 185

3.​2.​3 The Poisson Summation Formula 187

Exercises 191

3.​3 Decay of Fourier Coefficients 192

3.​3.​1 Decay of Fourier Coefficients of Arbitrary Integrable Functions 193

3.​3.​2 Decay of Fourier Coefficients of Smooth Functions 195

3.​3.​3 Functions with Absolutely Summable Fourier Coefficients 200

Exercises 202

3.​4 Pointwise Convergence of Fourier Series 204

3.​4.​1 Pointwise Convergence of the Fejér Means 204

3.​4.​2 Almost Everywhere Convergence of the Fejér Means 207

3.​4.​3 Pointwise Divergence of the Dirichlet Means 210

3.​4.​4 Pointwise Convergence of the Dirichlet Means 212

Exercises 214

3.​5 A Tauberian theorem and Functions of Bounded Variation 216

3.​5.​1 A Tauberian theorem 216

3.​5.​2 The sine integral function 218

3.​5.​3 Further properties of functions of bounded variation 219

3.​5.​4 Gibbs phenomenon 221

Exercises 225

3.​6 Lacunary Series and Sidon Sets 226

3.​6.​1 Definition and Basic Properties of Lacunary Series 227

3.6.2 Equivalence of $$L^{p}$$ Norms of Lacunary Series 229

3.​6.​3 Sidon sets 235

Exercises 237

4 Topics on Fourier Series 241

4.​1 Convergence in Norm, Conjugate Function, and Bochner–Riesz Means 241

4.​1.​1 Equivalent Formulations of Convergence in Norm 242

4.1.2 The $$L^{p}$$ Boundedness of the Conjugate Function 246

4.​1.​3 Bochner–Riesz Summability 250

Exercises 253

4.​2 A.​ E.​ Divergence of Fourier Series and Bochner–Riesz means 255

4.​2.​1 Divergence of Fourier Series of Integrable Functions 255

4.​2.​2 Divergence of Bochner–Riesz Means of Integrable Functions 261

Exercises 270

4.​3 Multipliers, Transference, and Almost Everywhere Convergence 271

4.​3.​1 Multipliers on the Torus 271

4.​3.​2 Transference of Multipliers 275

4.​3.​3 Applications of Transference 280

4.​3.​4 Transference of Maximal Multipliers 281

4.​3.​5 Applications to Almost Everywhere Convergence 285

4.​3.​6 Almost Everywhere Convergence of Square Dirichlet Means 287

Exercises 289

4.​4 Applications to Geometry and Partial Differential Equations 292

4.​4.​1 The Isoperimetric Inequality 292

4.​4.​2 The Heat Equation with Periodic Boundary Condition 294

Exercises 298

4.​5 Applications to Number theory and Ergodic theory 299

4.​5.​1 Evaluation of the Riemann Zeta Function at even Natural numbers 299

4.​5.​2 Equidistributed sequences 302

4.​5.​3 The Number of Lattice Points inside a Ball 305

Exercises 308

5 Singular Integrals of Convolution Type 313

5.​1 The Hilbert Transform and the Riesz Transforms 313

5.​1.​1 Definition and Basic Properties of the Hilbert Transform 314

5.​1.​2 Connections with Analytic Functions 317

5.1.3 L p Boundedness of the Hilbert Transform 319

5.​1.​4 The Riesz Transforms 324

Exercises 329

5.​2 Homogeneous Singular Integrals and the Method of Rotations 333

5.​2.​1 Homogeneous Singular and Maximal Singular Integrals 333

5.2.2 L ² Boundedness of Homogeneous Singular Integrals 336

5.​2.​3 The Method of Rotations 339

5.​2.​4 Singular Integrals with Even Kernels 341

5.​2.​5 Maximal Singular Integrals with Even Kernels 347

Exercises 353

5.​3 The Calderón–Zygmund Decomposition and Singular Integrals 355

5.​3.​1 The Calderón–Zygmund Decomposition 355

5.​3.​2 General Singular Integrals 358

5.3.3 L r Boundedness Implies Weak Type (1, 1) Boundedness 359

5.​3.​4 Discussion on Maximal Singular Integrals 362

5.​3.​5 Boundedness for Maximal Singular Integrals Implies Weak Type (1, 1) Boundedness 366

Exercises 371

5.4 Sufficient Conditions for L p Boundedness 374

5.4.1 Sufficient Conditions for L p Boundedness of Singular Integrals 375

5.​4.​2 An Example 378

5.​4.​3 Necessity of the Cancellation Condition 379

5.4.4 Sufficient Conditions for L p Boundedness of Maximal Singular Integrals 380

Exercises 384

5.​5 Vector-Valued Inequalities 385

5.5.1 ℓ ² -Valued Extensions of Linear Operators 386

5.5.2 Applications and ℓ r -Valued Extensions of Linear Operators 390

5.​5.​3 General Banach-Valued Extensions 391

Exercises 398

5.​6 Vector-Valued Singular Integrals 401

5.​6.​1 Banach-Valued Singular Integral Operators 402

5.​6.​2 Applications 408

5.​6.​3 Vector-Valued Estimates for Maximal Functions 411

Exercises 414

6 Littlewood–Paley Theory and Multipliers 419

6.​1 Littlewood–Paley Theory 419

6.​1.​1 The Littlewood–Paley Theorem 420

6.​1.​2 Vector-Valued Analogues 426

6.1.3 L p Estimates for Square Functions Associated with Dyadic Sums 426

6.1.4 Lack of Orthogonality on L p 431

Exercises 434

6.​2 Two Multiplier Theorems 437

6.2.1 The Marcinkiewicz Multiplier Theorem on R 439

6.2.2 The Marcinkiewicz Multiplier Theorem on R 441

6.2.3 The Mihlin–Hörmander Multiplier Theorem on R n 445

Exercises 450

6.​3 Applications of Littlewood–Paley Theory 453

6.​3.​1 Estimates for Maximal Operators 453

6.​3.​2 Estimates for Singular Integrals with Rough Kernels 455

6.3.3 An Almost Orthogonality Principle on L p 459

Exercises 461

6.​4 The Haar System, Conditional Expectation, and Martingales 463

6.​4.​1 Conditional Expectation and Dyadic Martingale Differences 464

6.​4.​2 Relation Between Dyadic Martingale Differences and Haar Functions 465

6.​4.​3 The Dyadic Martingale Square Function 469

6.​4.​4 Almost Orthogonality Between the Littlewood–Paley Operators and the Dyadic Martingale Difference Operators 471

Exercises 474

6.​5 The Spherical Maximal Function 475

6.​5.​1 Introduction of the Spherical Maximal Function 475

6.​5.​2 The First Key Lemma 478

6.​5.​3 The Second Key Lemma 479

6.​5.​4 Completion of the Proof 481

Exercises 481

6.​6 Wavelets and Sampling 482

6.​6.​1 Some Preliminary Facts 483

6.​6.​2 Construction of a Nonsmooth Wavelet 485

6.​6.​3 Construction of a Smooth Wavelet 486

6.​6.​4 Sampling 490

Exercises 494

7 Weighted Inequalities 499

7.1 The A p Condition 499

7.1.1 Motivation for the A p Condition 500

7.1.2 Properties of A p Weights 503

Exercises 511

7.2 Reverse Hölder Inequality for A p Weights and Consequences 514

7.2.1 The Reverse Hölder Property of A p Weights 514

7.​2.​2 Consequences of the Reverse Hölder Property 518

Exercises 521

7.3 The A ∞ Condition 525

7.3.1 The Class of A ∞ Weights 525

7.3.2 Characterizations of A ∞ Weights 527

Exercises 530

7.​4 Weighted Norm Inequalities for Singular Integrals 532

7.​4.​1 Singular Integrals of Non Convolution type 532

7.​4.​2 A Good Lambda Estimate for Singular Integrals 533

7.​4.​3 Consequences of the Good Lambda Estimate 539

7.4.4 Necessity of the A p Condition 543

Exercises 545

7.5 Further Properties of A p Weights 546

7.​5.​1 Factorization of Weights 546

7.5.2 Extrapolation from Weighted Estimates on a Single $$L^{p:{0}}$$ 548

7.​5.​3 Weighted Inequalities Versus Vector-Valued Inequalities 554

Exercises 558

A Gamma and Beta Functions563

A.1 A Useful Formula563

A.2 Definitions of $$\varGamma (z)$$ and B ( z , w ) 563

A.3 Volume of the Unit Ball and Surface of the Unit Sphere565

A.4 Computation of Integrals Using Gamma Functions565

A.5 Meromorphic Extensions of B ( z , w ) and $$\varGamma (z)$$ 566

A.6 Asymptotics of $$\varGamma (x)$$ as $$x \rightarrow \infty $$ 567

A.7 Euler’s Limit Formula for the Gamma Function568

A.8 Reflection and Duplication Formulas for the Gamma Function570

B Bessel Functions573

B.1 Definition573

B.2 Some Basic Properties573

B.3 An Interesting Identity576

B.4 The Fourier Transform of Surface Measure on $$\mathbf{S}^{n-1}$$ 577

B.5 The Fourier Transform of a Radial Function on $$\mathbf{R}^{n}$$ 577

B.6 Bessel Functions of Small Arguments578

B.7 Bessel Functions of Large Arguments579

B.8 Asymptotics of Bessel Functions580

B.9 Bessel Functions of general complex indices582

C Rademacher Functions585

C.1 Definition of the Rademacher Functions585

C.2 Khintchine’s Inequalities586

C.3 Derivation of Khintchine’s Inequalities586

C.4 Khintchine’s Inequalities for Weak Type Spaces589

C.5 Extension to Several Variables589

D Spherical Coordinates591

D.1 Spherical Coordinate Formula591

D.2 A Useful Change of Variables Formula592

D.3 Computation of an Integral over the Sphere593

D.4 The Computation of Another Integral over the Sphere593

D.5 Integration over a General Surface594

D.6 The Stereographic Projection594

E Some Trigonometric Identities and Inequalities597

F Summation by Parts599

G Basic Functional Analysis601

H The Minimax Lemma603

I Taylor’s and Mean Value Theorem in Several Variables607

I.1 Mutlivariable Taylor’s Theorem607

I.2 The Mean value Theorem608

J The Whitney Decomposition of Open Sets in $$\mathbf{R}^{n}$$ 609

J.1 Decomposition of Open Sets609

J.2 Partition of Unity adapted to Whitney cubes611

Glossary613

References617

Index633

© Springer Science+Business Media New York 2014

Loukas GrafakosClassical Fourier AnalysisGraduate Texts in Mathematics24910.1007/978-1-4939-1194-3_1

1.  L p Spaces and Interpolation

Loukas Grafakos1 

(1)

Department of Mathematics, University of Missouri, Columbia, MO, USA

Many quantitative properties of functions are expressed in terms of their integrability to a power. For this reason it is desirable to acquire a good understanding of spaces of functions whose modulus to a power p is integrable. These are called Lebesgue spaces and are denoted by $$L^{p}$$. Although an in-depth study of Lebesgue spaces falls outside the scope of this book, it seems appropriate to devote a chapter to reviewing some of their fundamental properties.

The emphasis of this review is basic interpolation between Lebesgue spaces. Many problems in Fourier analysis concern boundedness of operators on Lebesgue spaces, and interpolation provides a framework that often simplifies this study. For instance, in order to show that a linear operator maps L p to itself for all $$1 < p < \infty $$, it is sufficient to show that it maps the (smaller) Lorentz space L p, 1 into the (larger) Lorentz space $$L^{p,\infty }$$ for the same range of p’s. Moreover, some further reductions can be made in terms of the Lorentz space L p, 1. This and other considerations indicate that interpolation is a powerful tool in the study of boundedness of operators.

Although we are mainly concerned with L p subspaces of Euclidean spaces, we discuss in this chapter L p spaces of arbitrary measure spaces, since they represent a useful general setting. Many results in the text require working with general measures instead of Lebesgue measure.

1.1 L p and Weak L p

A measure space is a set X equipped with a $$\sigma$$-algebra of subsets of it and a function μ from the $$\sigma$$-algebra to $$[0,\infty ]$$ that satisfies $$\mu (\varnothing ) = 0$$ and

for any sequence B j of pairwise disjoint elements of the $$\sigma$$-algebra. The function μ is called a (positive) measure on X and elements of the $$\sigma$$-algebra of X are called measurable sets . Measure spaces will be assumed to be complete, i.e., subsets of the $$\sigma$$-algebra of measure zero also belong to the $$\sigma$$-algebra. A measure space X is called $$\sigma$$ -finite if there is a sequence of measurable subsets X n of it such that

and $$\mu (X_{n}) < \infty $$. A real-valued function f on a measure space is called measurable if the set $$\{x \in X:\,\, f(x) >\lambda \}$$ is measurable for all real numbers $$\lambda$$. A complex-valued function is measurable if and only if its real and imaginary parts are measurable. A simple function is a finite linear combination of characteristic functions of measurable subsets of X; these subsets may have infinite measure. A finitely simple function has the form

where $$N < \infty $$, $$c_{j} \in \mathbf{C}$$, and B j are pairwise disjoint measurable sets with $$\mu (B_{j}) < \infty $$. If $$N = \infty $$, this function will be called countably simple . Finitely simple functions are exactly the integrable simple functions. Every nonnegative measurable function is the pointwise limit of an increasing sequence of simple functions; if the space is σ-finite, these simple functions can be chosen to be finitely simple.

For 0 < p < ∞, L p (X, μ) denotes the set of all complex-valued μ-measurable functions on X whose modulus to the pth power is integrable. L ∞ (X, μ) is the set of all complex-valued μ-measurable functions f on X such that for some B > 0, the set {x:  | f(x) |  > B} has μ-measure zero. Two functions in L p (X, μ) are considered equal if they are equal μ-almost everywhere. When $$0 < p < \infty $$ finitely simple functions are dense in L p (X, μ). Within context and in the absence of ambiguity, L p (X, μ) is simply written as L p .

The notation $$L^{p}(\mathbf{R}^{n})$$ is reserved for the space $$L^{p}(\mathbf{R}^{n},\vert \cdot \vert )$$, where | ⋅ | denotes n-dimensional Lebesgue measure. Lebesgue measure on $$\mathbf{R}^{n}$$ is also denoted by dx. Other measures will be considered on the Borel $$\sigma$$ -algebra of $$\mathbf{R}^{n}$$, i.e., is the smallest $$\sigma$$-algebra that contains the closed subsets of $$\mathbf{R}^{n}$$. Measures on the $$\sigma$$-algebra of Borel measurable subsets are called Borel measures ; such measures will be assumed to be finite on compact subsets of $$\mathbf{R}^{n}$$. A Borel measure $$\mu$$ with $$\mu (\mathbf{R}^{n}) < \infty $$ is called a finite Borel measure . A Borel measure on $$\mathbf{R}^{n}$$ is called regular for all Borel measurable sets E we have

The space $$L^{p}(\mathbf{Z})$$ equipped with counting measure is denoted by $$\ell^{p}(\mathbf{Z})$$ or simply ℓ p .

For 0 < p < ∞, we define the L p norm of a function f (or quasi-norm if p < 1) by

(1.1.1)

and for p = ∞ by

(1.1.2)

It is well known that Minkowski’s (or the triangle) inequality

(1.1.3)

holds for all f, g in $$L^{p} = L^{p}(X,\mu )$$, whenever 1 ≤ p ≤ ∞. Since in addition $$\|f\|_{L^{p}(X,\mu )} = 0$$ implies that f = 0 (μ-a.e.), the L p spaces are normed linear spaces for 1 ≤ p ≤ ∞. For 0 < p < 1, inequality (1.1.3) is reversed when f, g ≥ 0. However, the following substitute of (1.1.3) holds:

(1.1.4)

and thus L p (X, μ) is a quasi-normed linear space. See also Exercise 1.1.5. For all 0 < p ≤ ∞, it can be shown that every Cauchy sequence in L p (X, μ) is convergent, and hence the spaces L p (X, μ) are complete. For the case 0 < p < 1 we refer to Exercise 1.1.8. Therefore, the L p spaces are Banach spaces for 1 ≤ p ≤ ∞ and quasi-Banach spaces for 0 < p < 1. For any $$p \in (0,\infty )\setminus \{1\}$$ we use the notation $$p' = \frac{p} {p-1}$$. Moreover, we set $$1' = \infty $$ and $$\infty ' = 1$$, so that p″ = p for all $$p \in (0,\infty ]$$. Hölder’s inequality says that for all $$p \in [1,\infty ]$$ and all measurable functions f, g on (X, μ) we have

It is a well-known fact that the dual (L p )∗ of L p is isometric to L p′ for all $$1 \leq p < \infty $$. Furthermore, the $$L^{p}$$ norm of a function can be obtained via duality when $$1 \leq p \leq \infty $$ as follows:

For the endpoint cases p = 1, $$p = \infty $$, see Exercise 1.4.12 (a), (b).

1.1.1 The Distribution Function

Definition 1.1.1.

For f a measurable function on X, the distribution function of f is the function d f defined on [0, ∞) as follows:

(1.1.5)

The distribution function d f provides information about the size of f but not about the behavior of f itself near any given point. For instance, a function on $$\mathbf{R}^{n}$$ and each of its translates have the same distribution function. It follows from Definition 1.1.1 that d f is a decreasing function of $$\alpha$$ (not necessarily strictly).

Fig. 1.1

The graph of a simple function $$f\! =\!\sum _{ k=1}^{3}a_{k}\chi _{E_{k}}$$ and its distribution function d f (α). Here $$B_{j}\! =\!\sum _{ k=1}^{j}\mu (E_{k})$$.

Example 1.1.2.

For pedagogical reasons we compute the distribution function d f of a nonnegative simple function

where the sets E j are pairwise disjoint and $$a_{1} >\ldots > a_{N} > 0$$. If α ≥ a ₁, then clearly d f (α) = 0. However, if $$a_{2} \leq \alpha < a_{1}$$ then | f(x) |  > α precisely when x ∈ E ₁, and in general, if $$a_{j+1} \leq \alpha < a_{j}$$, then | f(x) |  > α precisely when $$x \in E_{1} \cup \ldots \cup E_{j}$$. Setting

for $$j \in \{ 1,\ldots,N\}$$, $$B_{0} = a_{N+1} = 0$$, and $$a_{0} = \infty $$, we have

Note that these formulas are valid even when $$\mu (E_{i}) = \infty $$ for some i. Figure 1.1 presents an illustration of this example when N = 3 and $$\mu (E_{j}) < \infty $$ for all j.

Proposition 1.1.3.

Let f and g be measurable functions on (X,μ). Then for all α,β > 0 we have

1.

|g|≤|f| μ-a.e. implies that $$d_{g} \leq d_{f}$$ ;

2.

$$d_{cf}(\alpha ) = d_{f}(\alpha /\vert c\vert )$$ , for all $$c \in \mathbf{C}\setminus \{0\}$$ ;

3.

$$d_{f+g}(\alpha +\beta ) \leq d_{f}(\alpha ) + d_{g}(\beta )$$ ;

4.

$$d_{fg}(\alpha \beta ) \leq d_{f}(\alpha ) + d_{g}(\beta )$$ .

Proof.

The simple proofs are left to the reader. □ 

Knowledge of the distribution function d f provides sufficient information to evaluate the L p norm of a function f precisely. We state and prove the following important description of the L p norm in terms of the distribution function.

Proposition 1.1.4.

Let (X,μ) be a $$\sigma$$ -finite measure space. Then for f in $$L^{p}(X,\mu )$$, $$0 < p < \infty $$ , we have

(1.1.6)

Moreover, for any increasing continuously differentiable function $$\varphi$$ on $$[0,\infty )$$ with $$\varphi (0) = 0$$ and every measurable function f on X with $$\varphi (\vert f\vert )$$ integrable on X, we have

(1.1.7)

Proof.

Indeed, we have

where in the second equality we used Fubini’s theorem, which requires the measure space to be $$\sigma$$-finite. This proves (1.1.6). Identity (1.1.7) follows similarly, replacing the function $$\alpha ^{p}$$ by the more general function $$\varphi (\alpha )$$ which has similar properties. □ 

Definition 1.1.5.

For 0 < p < ∞, the space weak $$L^{p}(X,\mu )$$ is defined as the set of all μ-measurable functions f such that

(1.1.8)

(1.1.9)

is finite. The space weak L ∞ (X, μ) is by definition L ∞ (X, μ).

One should check that (1.1.9) and (1.1.8) are in fact equal. The weak L p spaces are denoted by $$L^{p,\infty }(X,\mu )$$. Two functions in L p, ∞ (X, μ) are considered equal if they are equal μ-a.e. The notation $$L^{p,\infty }(\mathbf{R}^{n})$$ is reserved for $$L^{p,\infty }(\mathbf{R}^{n},\vert \cdot \vert )$$. Using Proposition 1.1.3 (2), we can easily show that

(1.1.10)

for any complex constant k. The analogue of (1.1.3) is

(1.1.11)

where $$c_{p} =\max (2,2^{1/p})$$, a fact that follows from Proposition 1.1.3 (3), taking both $$\alpha$$ and $$\beta$$ equal to α∕2. We also have that

(1.1.12)

In view of (1.1.10), (1.1.11), and (1.1.12), $$L^{p,\infty }$$ is a quasi-normed linear space for 0 < p < ∞.

The weak L p spaces are larger than the usual L p spaces. We have the following:

Proposition 1.1.6.

For any 0 < p < ∞ and any f in L p (X,μ) we have

Hence the embedding $$L^{p}(X,\mu ) \subseteq L^{p,\infty }(X,\mu )$$ holds.

Proof.

This is just a trivial consequence of Chebyshev’s inequality :

Using (1.1.9) we obtain that $$\|f\|_{L^{p,\infty }} \leq \| f\|_{L^{p}}$$. □ 

The inclusion $$L^{p} \subseteq L^{p,\infty }$$ is strict. For example, on R n with the usual Lebesgue measure, let $$h(x) = \vert x\vert ^{-\frac{n} {p} }$$. Obviously, h is not in $$L^{p}(\mathbf{R}^{n})$$ but h is in $$L^{p,\infty }(\mathbf{R}^{n})$$ with $$\|h\|_{L^{p,\infty }(\mathbf{R}^{n})} = v_{n}^{1/p}$$, where v n is the measure of the unit ball of $$\mathbf{R}^{n}$$.

It is not immediate from their definition that the weak L p spaces are complete with respect to the quasi-norm $$\|\cdot \|_{L^{p,\infty }}$$. The completeness of these spaces is proved in Theorem 1.4.11, but it is also a consequence of Theorem 1.1.13, proved in this section.

1.1.2 Convergence in Measure

Next we discuss some convergence notions. The following notion is important in probability theory.

Definition 1.1.7.

Let f, f n , $$n = 1,2,\mathop{\ldots },$$ be measurable functions on the measure space (X, μ). The sequence f n is said to converge in measure to f if for all $$\varepsilon > 0$$ there exists an $$n_{0} \in \mathbf{Z}^{+}$$ such that

(1.1.13)

Remark 1.1.8.

The preceding definition is equivalent to the following statement:

(1.1.14)

Clearly (1.1.14) implies (1.1.13). To see the converse given $$\varepsilon > 0$$, pick $$0 <\delta <\varepsilon$$ and apply (1.1.13) for this δ. There exists an $$n_{0} \in \mathbf{Z}^{+}$$ such that

holds for n > n ₀. Since

we conclude that

for all n > n ₀. Let n → ∞ to deduce that

(1.1.15)

Since (1.1.15) holds for all $$0 <\delta <\varepsilon$$, (1.1.14) follows by letting δ → 0.

Convergence in measure is a weaker notion than convergence in either L p or $$L^{p,\infty }$$, $$0 < p \leq \infty $$, as the following proposition indicates:

Proposition 1.1.9.

Let 0 < p ≤∞ and f n , f be in L p,∞ (X,μ).

1.

If f n , f are in L p and f n → f in L p , then f n → f in $$L^{p,\infty }$$ .

2.

If $$f_{n} \rightarrow f$$ in $$L^{p,\infty }$$ , then f n converges to f in measure.

Proof.

Fix $$0 < p < \infty $$. Proposition 1.1.6 gives that for all $$\varepsilon > 0$$ we have

This shows that convergence in L p implies convergence in weak L p . The case $$p = \infty $$ is tautological.

Given $$\varepsilon > 0$$ find an n ₀ such that for n > n ₀, we have

Taking $$\alpha =\varepsilon$$, we conclude that convergence in L p, ∞ implies convergence in measure. □ 

Example 1.1.10.

Note that there is no general converse of statement (2) in the preceding proposition. Fix 0 < p < ∞ and on [0, 1] define the functions

Consider the sequence $$\{f_{1,1},\,f_{2,1},f_{2,2},\,f_{3,1},\,f_{3,2},\,f_{3,3},\ldots \}.$$ Observe that

Therefore, f k, j converges to 0 in measure. Likewise, observe that

which implies that f k, j does not converge to 0 in L p, ∞ .

It turns out that every sequence convergent in L p (X, μ) or in L p, ∞ (X, μ) has a subsequence that converges a.e. to the same limit.

Theorem 1.1.11.

Let f n and f be complex-valued measurable functions on a measure space (X,μ) and suppose that f n converges to f in measure. Then some subsequence of f n converges to f μ-a.e.

Proof.

For all $$k = 1,2,\ldots$$ choose inductively n k such that

(1.1.16)

and such that $$n_{1} < n_{2} < \cdots < n_{k} < \cdots $$. Define the sets

Equation (1.1.16) implies that

(1.1.17)

for all $$m = 1,2,3,\ldots$$. It follows from (1.1.17) that

(1.1.18)

Using (1.1.17) and (1.1.18), we conclude that the sequence of the measures of the sets $$\{\bigcup _{k=m}^{\infty }A_{k}\}_{m=1}^{\infty }$$ converges as $$m \rightarrow \infty $$ to

(1.1.19)

To finish the proof, observe that the null set in (1.1.19) contains the set of all x ∈ X for which $$f_{n_{k}}(x)$$ does not converge to f(x). □ 

In many situations we are given a sequence of functions and we would like to extract a convergent subsequence. One way to achieve this is via the next theorem, which is a useful variant of Theorem 1.1.11. We first give a relevant definition.

Definition 1.1.12.

We say that a sequence of measurable functions {f n } on the measure space (X, μ) is Cauchy in measure if for every $$\varepsilon > 0$$, there exists an $$n_{0} \in \mathbf{Z}^{+}$$ such that for n, m > n ₀ we have

Theorem 1.1.13.

Let (X,μ) be a measure space and let {f n } be a complex-valued sequence on X that is Cauchy in measure. Then some subsequence of f n converges μ-a.e.

Proof.

The proof is very similar to that of Theorem 1.1.11. For all $$k = 1,2,\ldots$$ choose n k inductively such that

(1.1.20)

and such that $$n_{1} < n_{2} <\ldots < n_{k} < n_{k+1} < \cdots $$. Define

As shown in the proof of Theorem 1.1.11, (1.1.20) implies that

(1.1.21)

For $$x\notin \bigcup _{k=m}^{\infty }A_{k}$$ and $$i \geq j \geq j_{0} \geq m$$ (and j ₀ large enough) we have

This implies that the sequence $$\{f_{n_{i}}(x)\}_{i}$$ is Cauchy for every x in the set $$(\bigcup _{k=m}^{\infty }A_{k})^{c}$$ and therefore converges for all such x. We define a function

Then $$f_{n_{j}} \rightarrow f$$ almost everywhere. □ 

1.1.3 A First Glimpse at Interpolation

It is a useful fact that if a function f is in L p (X, μ) and in L q (X, μ), then it also lies in L r (X, μ) for all p < r < q. The usefulness of the spaces L p, ∞ can be seen from the following sharpening of this statement:

Proposition 1.1.14.

Let 0 < p < q ≤∞ and let f in $$L^{p,\infty }(X,\mu ) \cap L^{q,\infty }(X,\mu )$$ , where X is a $$\sigma$$ -finite measure space. Then f is in L r (X,μ) for all p < r < q and

(1.1.22)

with the interpretation that $$1/\infty = 0$$ .

Proof.

Let us take first $$q < \infty $$. We know that

(1.1.23)

Set

(1.1.24)

We now estimate the L r norm of f. By (1.1.23), (1.1.24), and Proposition 1.1.4 we have

(1.1.25)

Observe that the integrals converge, since r − p > 0 and r − q < 0.

The case $$q = \infty $$ is easier. Since $$d_{f}(\alpha ) = 0$$ for $$\alpha >\| f\|_{L^{\infty }}$$ we need to use only the inequality $$d_{f}(\alpha ) \leq \alpha ^{-p}\|f\|_{L^{p,\infty }}^{p}$$ for $$\alpha \leq \| f\|_{L^{\infty }}$$ in estimating the first integral in (1.1.25). We obtain

which is nothing other than (1.1.22) when $$q = \infty $$. This completes the proof. □ 

Note that (1.1.22) holds with constant 1 if $$L^{p,\infty }$$ and $$L^{q,\infty }$$ are replaced by $$L^{p}$$ and $$L^{q}$$, respectively. It is often convenient to work with functions that are only locally in some $$L^{p}$$ space. This leads to the following definition.

Definition 1.1.15.

For 0 < p < ∞, the space $$L_{\mbox{ loc}}^{p}(\mathbf{R}^{n},\vert \cdot \vert )$$ or simply $$L_{\mbox{ loc}}^{p}(\mathbf{R}^{n})$$ is the set of all Lebesgue-measurable functions f on $$\mathbf{R}^{n}$$ that satisfy

(1.1.26)

for any compact subset K of $$\mathbf{R}^{n}$$. Functions that satisfy (1.1.26) with p = 1 are called locally integrable functions on $$\mathbf{R}^{n}$$.

The union of all $$L^{p}(\mathbf{R}^{n})$$ spaces for $$1 \leq p \leq \infty $$ is contained in $$L_{\mbox{ loc}}^{1}(\mathbf{R}^{n})$$. More generally, for 0 < p < q < ∞ we have the following:

Functions in $$L^{p}(\mathbf{R}^{n})$$ for 0 < p < 1 may not be locally integrable. For example, take $$f(x) = \vert x\vert ^{-n-\alpha }\chi _{\vert x\vert \leq 1},$$ which is in $$L^{p}(\mathbf{R}^{n})$$ when $$\alpha > 0$$ and $$p < n/(n+\alpha ),$$ and observe that $$f$$ is not integrable over any open set in $$\mathbf{R}^{n}$$ containing the origin.

Exercises

1.1.1. Suppose f and f n are measurable functions on (X, μ). Prove that

(a) d f is right continuous on [0, ∞).

(b) If $$\vert f\vert \leq \liminf _{n\rightarrow \infty }\vert f_{n}\vert $$ μ-a.e., then $$d_{f} \leq \liminf _{n\rightarrow \infty }d_{f_{n}}$$.

(c) If $$\vert f_{n}\vert \uparrow \vert f\vert $$, then $$d_{f_{n}} \uparrow d_{f}$$.

[Hint: Part (a): Let t n be a decreasing sequence of positive numbers that tends to zero. Show that $$d_{f}(\alpha _{0} + t_{n}) \uparrow d_{f}(\alpha _{0})$$ using a convergence theorem. Part (b): Let $$E =\{ x \in X:\,\, \vert f(x)\vert >\alpha \}$$ and $$E_{n} =\{ x \in X:\,\, \vert f_{n}(x)\vert >\alpha \}$$. Use that $$\mu \big(\bigcap _{n=m}^{\infty }E_{n}\big) \leq \liminf \limits _{n\rightarrow \infty }\mu (E_{n})$$ and $$E \subseteq \bigcup _{m=1}^{\infty }\bigcap _{n=m}^{\infty }E_{n}$$ μ-a.e.]

1.1.2. (Hölder’s inequality) Let $$0 < p,p_{1},\ldots,p_{k} \leq \infty $$, where $$k \geq 2$$, and let f j be in $$L^{p_{j}} = L^{p_{j}}(X,\mu )$$. Assume that

(a) Show that the product f ₁⋯f k is in L p and that

(b) When no p j is infinite, show that if equality holds in part (a), then it must be the case that $$c_{1}\vert f_{1}\vert ^{p_{1}} =\ldots = c_{k}\vert f_{k}\vert ^{p_{k}}$$ μ-a.e. for some c j  ≥ 0.

(c) Let 0 < q < 1 and $$q' = \frac{q} {q-1}$$. For r < 0 and g > 0 almost everywhere, define $$\|g\|_{L^{r}} =\| g^{-1}\|_{L^{\vert r\vert }}^{-1}$$. Show that if g is strictly positive μ-a.e. and lies in L q′ and f is measurable such that fg belongs to L ¹, we have

1.1.3. Let (X, μ) be a measure space.

(a) If f is in $$L^{p_{0}}(X,\mu )$$ for some p ₀ < ∞, prove that

(b) (Jensen’s inequality) Suppose that μ(X) = 1. Show that

for all $$0 < p < \infty $$.

(c) If μ(X) = 1 and f is in some $$L^{p_{0}}(X,\mu )$$ for some p ₀ > 0, then

with the interpretation $$e^{-\infty } = 0$$.

[Hint: Part (a): If $$0 <\| f\|_{L^{\infty }} < \infty $$, use that $$\|f\|_{L^{p}} \leq \| f\|_{L^{\infty }}^{(p-p_{0})/p}\|f\|_{L^{p_{0} }}^{p_{0}/p}$$ to obtain $$\limsup _{p\rightarrow \infty }\|f\|_{L^{p}} \leq \| f\|_{L^{\infty }}$$. Conversely, let $$E_{\gamma } =\{ x \in X:\,\, \vert f(x)\vert >\gamma \| f\|_{L^{\infty }}\}$$ for $$\gamma$$ in (0, 1). Then $$\mu (E_{\gamma }) > 0$$, $$\|f\|_{L^{p_{0}}(E_{\gamma })} > 0$$, and $$\|f\|_{L^{p}} \geq \big (\gamma \|f\|_{L^{\infty }}\big)^{(p-p_{0})/p}\|f\|_{L^{p_{ 0}}(E_{\gamma })}^{p_{0}/p}$$, hence $$\liminf _{p\rightarrow \infty }\|f\|_{L^{p}} \geq \gamma \| f\|_{L^{\infty }}$$. If $$\|f\|_{L^{\infty }} = \infty $$, set $$G_{n} =\{ \vert f\vert > n\}$$ and use that $$\|f\|_{L^{p}} \geq \| f\|_{L^{p}(G_{n})} \geq n\mu (G_{n})^{\frac{1} {p} }$$ to obtain $$\liminf _{p\rightarrow \infty }\|f\|_{L^{p}} \geq n$$. Part (b) is a direct consequence of Jensen’s inequality $$\int _{X}\log \vert h\vert \,d\mu \leq \log \big (\int _{X}\vert h\vert \,d\mu \big)$$. Part (c): Fix a sequence $$0 < p_{n} < p_{0}$$ such that $$p_{n} \downarrow 0$$ and define

Use that $$\frac{1} {p}(t^{p} - 1) \downarrow \log t$$ as p ↓ 0 for all t > 0. The Lebesgue monotone convergence theorem yields $$\int _{X}h_{n}\,d\mu \uparrow \int _{X}h\,d\mu$$, hence $$\int _{X} \frac{1} {p_{n}}(\vert f\vert ^{p_{n}} - 1)\,d\mu \downarrow \int _{X}\log \vert f\vert \,d\mu$$, where the latter could be $$-\infty $$. Use

to complete the proof. ]

1.1.4. Let $$a_{j}$$ be a sequence of positive reals. Show that

(a) $$\big(\sum _{j=1}^{\infty }a_{j}\big)^{\theta } \leq \sum _{j=1}^{\infty }a_{j}^{\theta },$$ for any 0 ≤ θ ≤ 1.

(b) $$\sum _{j=1}^{\infty }a_{j}^{\theta } \leq \big (\sum _{j=1}^{\infty }a_{j}\big)^{\theta },$$ for any $$1 \leq \theta < \infty $$.

(c) $$\big(\sum _{j=1}^{N}a_{j}\big)^{\theta } \leq N^{\theta -1}\sum _{j=1}^{N}a_{j}^{\theta }$$, when 1 ≤ θ < ∞.

(d) $$\sum _{j=1}^{N}a_{j}^{\theta } \leq N^{1-\theta }\big(\sum _{j=1}^{N}a_{j}\big)^{\theta }$$, when 0 ≤ θ ≤ 1.

1.1.5. Let $$\{f_{j}\}_{j=1}^{N}$$ be a sequence of L p (X, μ) functions.

(a) (Minkowski’s inequality) For 1 ≤ p ≤ ∞ show that

(b) (Reverse Minkowski inequality) For 0 < p < 1 and f j  ≥ 0 prove that

(c) For 0 < p < 1 show that

(d) The constant $$N^{\frac{1-p} {p} }$$ in part (c) is best possible.

[Hint: Part (c): Use Exercise 1.1.4 (c). Part (d): Take $$\{f_{j}\}_{j=1}^{N}$$ to be characteristic functions of disjoint sets with the same measure.]

1.1.6. (a) (Minkowski’s integral inequality) Let (X, μ) and (T, ν) be two $$\sigma$$-finite measure spaces and let 1 ≤ p < ∞. Show that for every nonnegative measurable function $$F$$ on the product space (X, μ) × (T, ν) we have

(b) State and prove an analogous inequality when $$p = \infty $$.

(c) Prove that when 0 < p < 1, then the preceding inequality is reversed.

(d) (Y. Sawano) Consider the example X = T = [0, 1], μ is counting measure, ν is Lebesgue measure, F(x, t) = 1 when x = t and zero otherwise. What is the relevance of this example with the inequalities in (a) and (b)?

[Hint: Part (a) Split the power p as 1 + (p − 1) and apply Hölder’s inequality with exponents p and p′. Part (b) Let $$p \rightarrow \infty $$ on subsets of X with finite measure.]

1.1.7. Let $$f_{1},\ldots,f_{N}$$ be in $$L^{p,\infty }(X,\mu )$$.

(a) Prove that for 1 ≤ p < ∞ we have

(b) Show that for 0 < p < 1 we have

[Hint: Use that $$\mu (\{\vert f_{1} +\ldots +f_{N}\vert >\alpha \}) \leq \sum _{j=1}^{N}\mu (\{\vert f_{j}\vert >\alpha /N\})$$ and Exercise 1.1.4 (a) and (c).]

1.1.8. Let 0 < p < ∞. Prove that L p (X, μ) is a complete quasi-normed space. This means that every quasi-norm Cauchy sequence is quasi-norm convergent.

[Hint: Let f n be a Cauchy sequence in L p . Pass to a subsequence $$\{n_{i}\}_{i}$$ such that $$\|f_{n_{i+1}} - f_{n_{i}}\|_{L^{p}} \leq 2^{-i}$$. Then the series $$f = f_{n_{1}} +\sum _{ i=1}^{\infty }(f_{n_{i+1}} - f_{n_{i}})$$ converges in $$L^{p}$$.]

1.1.9. Let (X, μ) be a measure space with μ(X) < ∞. Suppose that a sequence of measurable functions f n on X converges to f μ-a.e. Prove that f n converges to f in measure.

[Hint: For $$\varepsilon > 0$$, $$\big\{x \in X:\,\, f_{n}(x) \rightarrow f(x)\} \subseteq \bigcup \limits _{m=1}^{\infty }\bigcap \limits _{n=m}^{\infty }\{x \in X:\,\, \vert f_{n}(x) - f(x)\vert <\varepsilon \big\}.$$]

1.1.10. Let f be a measurable function on (X, μ) such such $$d_{f}(\alpha ) < \infty $$ for all $$\alpha > 0$$. Fix γ > 0 and define $$f_{\gamma } = f\chi _{\vert f\vert >\gamma }$$ and $$f^{\gamma } = f - f_{\gamma } = f\chi _{\vert f\vert \leq \gamma }$$.

(a) Prove that

(b) If $$f \in L^{p}(X,\mu )$$ then

(c) If f is in L p, ∞ (X, μ) prove that f γ is in L q (X, μ) for any q > p and f γ is in L q (X, μ) for any q < p. Thus $$L^{p,\infty }\subseteq L^{p_{0}} + L^{p_{1}}$$ when $$0 < p_{0} < p < p_{1} \leq \infty $$.

1.1.11. Let (X, μ) be a measure space and let E be a subset of X with μ(E) < ∞. Assume that f is in L p, ∞ (X, μ) for some 0 < p < ∞.

(a) Show that for 0 < q < p we have

(b) Conclude that if μ(X) < ∞ and 0 < q < p, then

[Hint: Part (a): Use $$\mu \big(E \cap \{\vert f\vert >\alpha \}\big) \leq \min \big (\mu (E),\alpha ^{-p}\big\|f\big\|_{L^{p,\infty }}^{p}\big).$$]

1.1.12. (Normability of weak $$L^{p}$$ for p > 1) Let (X, μ) be a $$\sigma$$-finite measure space and let $$0 < p < \infty $$. Pick 0 < r < p and define

where the supremum is taken over all measurable subsets E of X of finite measure.

(a) Use Exercise 1.1.11 with q = r to conclude that

for all f in L p, ∞ (X, μ). (It is not needed that X be $$\sigma$$-finite here).

(b) Prove that for all f in $$L^{p,\infty }(X,\mu )$$ we have

(Y. Oi) Notice that if X = { 1, 2}, μ({1}) = 1, μ({2}) = ∞, then X is not $$\sigma$$-finite, and verify that for the function f = 1 the preceding inequality fails.

(c) Show that $$L^{p,\infty }(X,\mu )$$ is metrizable for all $$0 < p < \infty $$, i.e., there is a metric on the space that generates the same topology as the quasi-norm. Also show that $$L^{p,\infty }(X,\mu )$$ is normable when p > 1, i.e., there is a norm on the space equivalent to $$\|\cdot \|_{L^{p,\infty }}$$.

(d) Use the characterization of the weak L p quasi-norm obtained in parts (a) and (b) to prove Fatou’s lemma for this space: For all measurable functions g n on X we have

for some constant C p that depends only on $$p \in (0,\infty )$$.

[Hint: Part (b): Write $$X =\bigcup _{ k=1}^{\infty }X_{k}$$ with $$\mu (X_{k}) < \infty $$ and take $$E =\{ \vert f\vert >\alpha \} \cap X_{k}$$.]

1.1.13. Consider the N! functions on the line

where σ is a permutation of the set $$\{1,2,\ldots,N\}$$.

(a) Show that each f σ satisfies $$\|f_{\sigma }\|_{L^{1,\infty }} = 1$$.

(b) Show that $$\|\sum _{\sigma \in S_{N}}f_{\sigma }\|_{L^{1,\infty }} = N!\big(1 + \frac{1} {2} +\ldots + \frac{1} {N}\big)$$.

(c) Conclude that the space $$L^{1,\infty }(\mathbf{R})$$ is not normable (this means that $$\|\cdot \|_{L^{1,\infty }}$$ is not equivalent to a norm).

(d) Use a similar argument to prove that $$L^{1,\infty }(\mathbf{R}^{n})$$ is not normable by considering the functions

where $$\sigma$$ is a permutation of the set $$\{1,2,\ldots,N^{n}\}$$ and τ is a fixed injective map from the set of all n-tuples of integers with coordinates 1 ≤ j ≤ N onto the set $$\{1,2,\ldots,N^{n}\}$$. One may take

for instance.

1.1.14. Let (X, μ) be a measure space and let s > 0.

(a) Let f be a measurable function on X. Show that if $$0 < p < q < \infty $$ we have

(b) Let f j , 1 ≤ j ≤ m, be measurable functions on X and let $$0 < p < \infty $$. Show that

(c) Conclude from part (b) that for 0 < p < 1 we have

The latter estimate is referred to as the p-normability of weak L p .

[Hint: Part (a): Use the distribution function. Part (c): First obtain the estimate

for all $$\alpha > 0$$ and then use part (b).]

1.1.15. (Hölder’s inequality for weak spaces) Let f j be in $$L^{p_{j},\infty }$$ of a measure space X where 0 < p j  < ∞ and 1 ≤ j ≤ k. Let

Prove that

[Hint: Take $$\|f_{j}\|_{L^{p_{j},\infty }} = 1$$ for all j. Control $$d_{f_{1}\cdots f_{k}}(\alpha )$$ by

Set $$x_{1} = s_{1}/\alpha$$, $$x_{2} = s_{2}/s_{1},\ldots,x_{k} = 1/s_{k-1}$$. Minimize $$x_{1}^{p_{1}} +\ldots +x_{k}^{p_{k}}$$ subject to the constraint $$x_{1}\cdots x_{k} = 1/\alpha$$.]

1.1.16. Let $$0 < p_{0} < p < p_{1} \leq \infty $$ and let $$\frac{1} {p} = \frac{1-\theta } {p_{0}} + \frac{\theta } {p_{ 1}}$$ for some θ ∈ [0, 1]. Prove the following:

1.1.17. ([231]) Follow the steps below to prove the isoperimetric inequality. For $$n \geq 2$$ and 1 ≤ j ≤ n define the projection maps $$\pi _{j}:\,\, \mathbf{R}^{n} \rightarrow \mathbf{R}^{n-1}$$ by setting for $$x = (x_{1},\ldots,x_{n})$$,

with the obvious interpretations when j = 1 or j = n.

(a) For maps $$f_{j}:\,\, \mathbf{R}^{n-1} \rightarrow \mathbf{C}$$ prove that

(b) Let $$\varOmega$$ be a compact set with a rectifiable boundary in $$\mathbf{R}^{n}$$ where $$n \geq 2$$. Show that there is a constant c n independent of $$\varOmega$$ such that

where the expression $$\vert \partial \varOmega \vert $$ denotes the (n​ −​ 1)-dimensional surface measure of the boundary of $$\varOmega$$.

[Hint: Part (a): Use induction starting with n = 2. For n ≥ 3 write

where $$P(x_{1},\ldots,x_{n-1}) =\int _{\mathbf{R}}\vert f_{1}(\pi _{1}(x))\cdots f_{n-1}(\pi _{n-1}(x))\vert \,dx_{n},$$ and apply the induction hypothesis to the n − 1 functions

for $$j = 1,\ldots,n - 1$$, to obtain the required conclusion. Part (b): Specialize part (a) to the case $$f_{j} =\chi _{\pi _{j}[\varOmega ]}$$ to obtain

and then use that $$\vert \pi _{j}[\varOmega ]\vert \leq \frac{1} {2}\vert \partial \varOmega \vert $$.]

1.2 Convolution and Approximate Identities

The notion of convolution can be defined on measure spaces endowed with a group structure. It turns out that the most natural environment to define convolution is the context of topological groups. Although the focus of this book is harmonic analysis on Euclidean spaces, we develop the notion of convolution on general groups. This allows us to study this concept on $$\mathbf{R}^{n}$$, $$\mathbf{Z}^{n}$$, and $$\mathbf{T}^{n}$$, in a unified way. Moreover, since the basic properties of convolutions and approximate identities do not require commutativity of the group operation, we may assume that the underlying groups are not necessarily abelian. Thus, the results in this section can be also applied to nonabelian structures such as the Heisenberg group.

1.2.1 Examples of Topological Groups

A topological group G is a Hausdorff topological space that is also a group with law

(1.2.1)

such that the maps (x, y)↦xy and x↦x ⁻¹ are continuous. The identity element of the group is the unique element e with the property xe = ex = x for all x ∈ G. We adopt the standard notation

for subsets A and B of G. Note that $$(AB)^{-1} = B^{-1}A^{-1}$$. Every topological group G has an open basis at e consisting of symmetric neighborhoods, i.e., open sets U satisfying U = U ⁻¹. A topological group is called locally compact if there is an open set U containing the identity element such that $$\overline{U}$$ is compact. Then every point in the group has an open neighborhood with compact closure.

Let G be a locally compact group. It is known that G possesses a positive measure $$\lambda$$ on the Borel sets that is nonzero on all nonempty open sets, finite on compact sets, and is left invariant, meaning that

(1.2.2)

for all measurable sets A and all t ∈ G. Such a measure $$\lambda$$ is called a (left) Haar measure on G. Similarly, G possesses a right Haar measure which is right invariant, i.e., $$\lambda (At) =\lambda (A)$$ for all measurable $$A \subseteq G$$ and all t ∈ G. For the existence of Haar measure we refer to [152, §15] or [213, §16.3]. Furthermore, Haar measure is unique up to positive multiplicative constants. If G is abelian then any left Haar measure on G is a constant multiple of any given right Haar measure on G. A locally compact group which is a countable union of compact subsets is a $$\sigma$$-finite measure space under left or right Haar measure. This is case for connected locally compact groups.

Example 1.2.1.

The standard examples are provided by the spaces $$\mathbf{R}^{n}$$ and $$\mathbf{Z}^{n}$$ with the usual topology and the usual addition of n-tuples. Another example is the space $$\mathbf{T}^{n} = \mathbf{R}^{n}/\mathbf{Z}^{n}$$ defined as follows:

with the usual topology and group law:

Example 1.2.2.

Let $$G = \mathbf{R}^{{\ast}} = \mathbf{R}\setminus \{0\}$$ with group law the usual multiplication. It is easy to verify that the measure $$\lambda = dx/\vert x\vert $$ is invariant under multiplicative translations, that is,

for all f in L ¹(G, μ) and all $$t \in \mathbf{R}^{{\ast}}$$. Therefore, dx∕ | x | is a Haar measure. [Taking f = χ A gives $$\lambda (tA) =\lambda (A)$$.]

Example 1.2.3.

Similarly, on the multiplicative group $$G = \mathbf{R}^{+}$$, a Haar measure is dx∕x.

Example 1.2.4.

Counting measure is a Haar measure on the group $$\mathbf{Z}^{n}$$ with the usual addition as group operation.

Example 1.2.5.

The Heisenberg group $$\mathbf{H}^{n}$$ is the set $$\mathbf{C}^{n} \times \mathbf{R}$$ with the group operation

It can easily be seen that the identity element e of this group is $$0 \in \mathbf{C}^{n} \times \mathbf{R}$$ and $$(z_{1},\ldots,z_{n},t)^{-1} = (-z_{1},\ldots,-z_{n},-t)$$. Topologically the Heisenberg group is identified with $$\mathbf{C}^{n} \times \mathbf{R}$$, and both left and right Haar measure on $$\mathbf{H}^{n}$$ is Lebesgue measure. The norm

introduces balls $$B_{r}(x) =\{ y \in \mathbf{H}^{n}:\, \vert y^{-1}x\vert < r\}$$ on the Heisenberg group that are quite different from Euclidean balls. For x close to the origin, the balls B r (x) are not far from being Euclidean, but for x far away from e = 0 they look like slanted truncated cylinders. The Heisenberg group can be naturally identified as the boundary of the unit ball in $$\mathbf{C}^{n}$$ and plays an important role in quantum mechanics.

1.2.2 Convolution

Throughout the rest of this section, we fix a locally compact group G and a left invariant Haar measure λ on G. We assume that G is a countable union of compact subsets, hence the pair $$(G,\lambda )$$ forms a $$\sigma$$-finite measure space. The spaces L p (G, λ) and $$L^{p,\infty }(G,\lambda )$$ are simply denoted by $$L^{p}(G)$$ and L p, ∞ (G).

Left invariance of λ is equivalent to the fact that for all t ∈ G and all nonnegative measurable functions f on G we have

(1.2.3)

Equation (1.2.3) is a restatement of (1.2.2) if f is a characteristic function. Obviously (1.2.3) also holds for $$f \in L^{1}(G)$$ by linearity and approximation.

We are now ready to define the operation of convolution.

Definition 1.2.6.

Let f, g be in L ¹(G). Define the convolution f ∗ g by

(1.2.4)

For instance, if $$G = \mathbf{R}^{n}$$ with the usual additive structure, then y ⁻¹ = −y and the integral in (1.2.4) is written as

Remark 1.2.7.

The right-hand side of (1.2.4) is defined a.e., since the following double integral converges absolutely:

The change of variables $$z = x^{-1}y$$ yields that (1.2.4) is in fact equal to

(1.2.5)

where the substitution of $$d\lambda (y)$$ by $$d\lambda (z)$$ is justified by left invariance.

Example 1.2.8.

On $$\mathbf{R}$$ let f(x) = 1 when − 1 ≤ x ≤ 1 and zero otherwise. We see that (f ∗ f)(x) is equal to the length of the intersection of the intervals [−1, 1] and [x − 1, x + 1]. It follows that (f ∗ f)(x) = 2 − | x | for | x | ≤ 2 and zero otherwise. Observe that f ∗ f is a smoother function than f. Similarly, we obtain that f ∗ f ∗ f is a smoother function than f ∗ f.

There is an analogous calculation when g is the characteristic function of the unit disk B(0, 1) in R ². A simple computation gives

when x = (x ₁, x ₂) in R ² satisfies | x | ≤ 2, while (g ∗ g)(x) = 0 if | x | ≥ 2.

A calculation similar to that in Remark 1.2.7 yields that

(1.2.6)

that is, the convolution of two integrable functions is also an integrable function with L ¹ norm less than or equal to the product of the L ¹ norms.

Proposition 1.2.9.

For all f, g, h in L ¹ (G), the following properties are valid:

1.

$$f {\ast} (g {\ast} h) = (f {\ast} g) {\ast} h$$ (associativity)

2.

f ∗ (g + h) = f ∗ g + f ∗ h and (f + g) ∗ h = f ∗ h + g ∗ h (distributivity)

Proof.

The easy proofs are omitted. □ 

Proposition 1.2.9 implies that L ¹(G) is a (not necessarily commutative) Banach algebra under the convolution product.

1.2.3 Basic Convolution Inequalities

The most fundamental inequality involving convolutions is the following.

Theorem 1.2.10.

(Minkowski’s inequality) Let 1 ≤ p ≤∞. For f in L p (G) and g in L ¹ (G) we have that g ∗ f exists $$\lambda$$ -a.e. and satisfies

(1.2.7)

Proof.

Estimate (1.2.7) follows directly from Exercise 1.1.6. Here we give a direct proof. We may assume that 1 < p < ∞, since the cases p = 1 and p = ∞ are simple. We first show that the convolution | g | ∗ | f | exists $$\lambda$$-a.e. Indeed,

(1.2.8)

Apply Hölder’s inequality in (1.2.8) with respect to the measure $$\vert g(y)\vert \,d\lambda (y)$$ to the functions y↦f(y ⁻¹ x) and 1 with exponents p and p′ = p∕(p − 1), respectively. We obtain

(1.2.9)

Taking L p norms of both sides of (1.2.9) we deduce

where the second equality follows by Fubini’s theorem. This shows that | g | ∗ | f | is finite $$\lambda$$-a.e. and satisfies (1.2.7); then g ∗ f exists $$\lambda$$-a.e. and also satisfies (1.2.7), since $$\vert g {\ast} f\vert \leq \vert g\vert {\ast}\vert f\vert $$. □ 

Remark 1.2.11.

Theorem 1.2.10 may fail for nonabelian groups if g ∗ f is replaced by f ∗ g in (1.2.7). Note, however, that if for all h ∈ L ¹(G) we have

(1.2.10)

where $$\widetilde{h}(x) = h(x^{-1})$$, then (1.2.7) holds when the quantity $$\|g {\ast} f\|_{L^{p}(G)}$$ is replaced by $$\|f {\ast} g\|_{L^{p}(G)}$$. To see this, observe that if (1.2.10) holds, then we can use (1.2.5) to conclude that if f in L p (G) and g in L ¹(G), then

(1.2.11)

If the left Haar measure satisfies

(1.2.12)

for all measurable $$A \subseteq G$$, then (1.2.10) holds and thus (1.2.11) is satisfied for all g in L ¹(G) and f ∈ L p (G). This is, for instance, the case for the Heisenberg group $$\mathbf{H}^{n}$$.

Minkowski’s inequality (1.2.11) is only a special case of Young’s inequality in which the function g can be in any space L r (G) for $$1 \leq r \leq \infty $$.

Theorem 1.2.12.

(Young’s inequality) Let 1 ≤ p,q,r ≤∞ satisfy

(1.2.13)

Then for all f in L p (G) and all g in L r (G) satisfying $$\big\|g\big\|_{L^{r}(G)} =\big\|\widetilde{ g}\,\big\|_{L^{r}(G)}$$ we have f ∗ g exists λ-a.e. and satisfies

(1.2.14)

Proof.

Young’s inequality is proved in a way similar to Minkowski’s inequality. We do a suitable splitting of the product | f(y) | | g(y ⁻¹ x) | and apply Hölder’s inequality. Observe that when $$r < \infty $$, the hypotheses on the indices imply that

Using Hölder’s inequality with exponents r′, q, and p′, we obtain

where we used left invariance. Now take L q norms (in x) and apply Fubini’s theorem to deduce that

using the hypothesis on g. This implies that | f | ∗ | g | is finite $$\lambda$$-a.e. and satisfies (1.2.14); then f ∗ g exists $$\lambda$$-a.e. and also satisfies (1.2.14).

Finally, note that if $$r = \infty $$, the assumptions on p and q imply that p = 1 and $$q = \infty $$, in which case the required inequality trivially holds. □ 

We now give a version of Theorem 1.2.12 for weak L p spaces. Theorem 1.2.13 is improved in Section 1.4.

Theorem 1.2.13.

(Young’s inequality for weak type spaces) Let G be a locally compact group with left Haar measure λ that satisfies (1.2.12) . Let 1 ≤ p < ∞ and 1 < q,r < ∞ satisfy

(1.2.15)

Then there exists a constant C p,q,r > 0 such that for all f in L p (G) and g in L r,∞ (G), the convolution f ∗ g exists $$\lambda$$ -a.e. and satisfies

(1.2.16)

Proof.

As in the proofs of Theorems 1.2.10 and 1.2.12, we first obtain (1.2.16) for the convolution of the absolute values of the functions. This implies that $$\vert f\vert {\ast}\vert g\vert < \infty \,\,$$ $$\lambda$$-a.e., and thus f ∗ g exists $$\lambda$$-a.e. and satisfies | f ∗ g | ≤ | f | ∗ | g | . We may therefore assume that f, g ≥ 0 $$\lambda$$-a.e. The proof is based on a suitable splitting of the function g. Let M be a positive real number to be chosen later. Define $$g_{1} = g\chi _{\vert g\vert \leq M}$$ and $$g_{2} = g\chi _{\vert g\vert >M}$$. In view of Exercise 1.1.10 (a) we have

(1.2.17)

(1.2.18)

Proposition 1.1.3 gives for all β > 0

(1.2.19)

and thus it suffices to estimate the distribution functions of f ∗ g ₁ and f ∗ g ₂. Since g ₁ is the “small” part of g, it is in L s for any s > r. In fact, we have

(1.2.20)

when $$s < \infty $$.

Similarly, since g ₂ is the “large” part of g, it is in L t for any t < r, and

(1.2.21)

Since 1∕r = 1∕p′ + 1∕q, it follows that 1 < r < p′. Select t = 1 and s = p′. Hölder’s inequality and (1.2.20) give when $$p' < \infty $$

(1.2.22)

and

(1.2.23)

when $$p' = \infty $$. If $$p' < \infty $$ choose an M such that the right-hand side of (1.2.22) is equal to $$\beta /2$$. If $$p' = \infty $$ choose M such that the right-hand side of (1.2.23) is also equal to $$\beta /2$$. That is, choose

if $$p' < \infty $$ and $$M =\beta /(2\|f\|_{L^{1}})$$ if $$p' = \infty $$. For these choices of M we have that

Next by Theorem 1.2.10 and (1.2.21) with t = 1 we obtain

(1.2.24)

For the value of M chosen,

using (1.2.24) and Chebyshev’s inequality, we obtain

(1.2.25)

which is the required inequality. This proof gives that the constant C p, q, r blows up like (r − 1)−p∕q as r → 1. □ 

Example 1.2.14.

Theorem 1.2.13 may fail at some endpoints:

(1)

r = 1 and $$1 \leq p = q \leq \infty $$. On $$\mathbf{R}$$ take g(x) = 1∕ | x | and f = χ [0, 1]. Clearly, g is in $$L^{1,\infty }$$ and f in L p for all $$1 \leq p \leq \infty $$, but the convolution of f and g is identically equal to infinity on the interval [0, 1]. Therefore, (1.2.16) fails in this case.

(2)

q = ∞ and $$1 < r = p' < \infty $$. On $$\mathbf{R}$$ let $$f(x) = (\vert x\vert ^{1/p}\log \vert x\vert )^{-1}$$ for | x | ≥ 2 and zero otherwise, and also let $$g(x) = \vert x\vert ^{-1/r}$$. We see that $$(f {\ast} g)(x) = \infty $$ for | x | ≤ 1. Thus (1.2.16) fails in this case also.

(3)

$$r = q = \infty $$ and p = 1. Then inequality (1.2.16) trivially holds.

1.2.4 Approximate Identities

We now introduce the notion of approximate identities. The Banach algebra L ¹(G) may not have a unit element, that is, an element f ₀ such that

(1.2.26)

for all f ∈ L ¹(G). In particular, this is the case when $$G = \mathbf{R}$$; in fact, the only f ₀ that satisfies (1.2.26) for all $$f \in L^{1}(\mathbf{R})$$ is not a function but the Dirac delta distribution, introduced in Chapter 2. It is reasonable therefore to introduce the notion of approximate unit or identity, a family of functions $$k_{\varepsilon }$$ with the property $$k_{\varepsilon } {\ast} f \rightarrow f$$ in L ¹ as $$\varepsilon \rightarrow 0$$.

Definition 1.2.15.

An approximate identity (as $$\varepsilon \rightarrow 0$$) is a family of L ¹(G) functions $$k_{\varepsilon }$$ with the following three properties:

(i)

   There exists a constant c > 0 such that $$\|k_{\varepsilon }\|_{L^{1}(G)} \leq c$$ for all $$\varepsilon > 0$$.

(ii)

  $$\int _{G}k_{\varepsilon }(x)\,d\lambda (x) = 1$$ for all $$\varepsilon > 0$$.

(iii)

 For any neighborhood V of the identity element e of the group G we have $$\int _{V ^{c}}\vert k_{\varepsilon }(x)\vert \,d\lambda (x) \rightarrow 0$$ as $$\varepsilon \rightarrow 0$$.

The construction of approximate identities on general locally compact groups G is beyond the scope of this book and is omitted; see [152] for details. In this book we are interested only in groups with Euclidean structure, where approximate identities exist in abundance.

Sometimes we think of approximate identities as sequences $$\{k_{n}\}_{n}$$. In this case property (iii) holds as n → ∞. It is best to visualize approximate identities as sequences of positive functions k n that spike near 0 in such a way that the signed area under the graph of each function remains constant (equal to one) but the support shrinks to zero. See Figure 1.2.

Example 1.2.16.

On $$\mathbf{R}$$ let $$P(x) = (\pi (x^{2} + 1))^{-1}$$ and $$P_{\varepsilon }(x) =\varepsilon ^{-1}P(\varepsilon ^{-1}x)$$ for $$\varepsilon > 0$$. Since $$P_{\varepsilon }$$ and P have the same L ¹ norm and

property (ii) is satisfied. Property (iii) follows from the fact that

for all δ > 0. The function $$P_{\varepsilon }$$ is called the Poisson kernel .

The Poisson kernel may be replaced by any integrable function of integral 1 as the following example indicates.

Example 1.2.17.

On $$\mathbf{R}^{n}$$ let k(x) be an integrable function with integral one. Let $$k_{\varepsilon }(x) =\varepsilon ^{-n}k(\varepsilon ^{-1}x)$$. It is straightforward to see that $$k_{\varepsilon }(x)$$ is an approximate identity. Property (iii) follows from the fact that

as $$\varepsilon \rightarrow 0$$ for $$\delta$$ fixed.

Fig. 1.2

The Fejér kernel F ₅ plotted on the interval $$[-\tfrac{1} {2}, \tfrac{1} {2}]$$.

Example 1.2.18.

On the circle group T ¹ let

(1.2.27)

To check the previous equality we use that

and we carry out the calculation. F N is called the Fejér kernel . See Figure 1.2. To see that the sequence $$\{F_{N}\}_{N}$$ is an approximate identity, we check conditions (i), (ii), and (iii) in Definition 1.2.15. Property (iii) follows from the expression giving F N in terms of sines, while property (i) follows from the expression giving F N in terms of exponentials. Property (ii) is identical to property (i), since F N is nonnegative.

Next comes the basic theorem concerning approximate identities.

Theorem 1.2.19.

Let $$k_{\varepsilon }$$ be an approximate identity on a locally compact group G with left Haar measure λ.

(1)

If f lies in L p (G) for 1 ≤ p < ∞, then $$\|k_{\varepsilon } {\ast} f - f\|_{L^{p}(G)} \rightarrow 0$$ as $$\varepsilon \rightarrow 0$$ .

(2)

Let f be a function in $$L^{\infty }(G)$$ that is uniformly continuous on a subset K of G, in the sense that for all $$\delta > 0$$ there is a neighborhood V of the identity element such that for all $$x \in K$$ and $$y \in V$$ we have $$\vert f(y^{-1}x) - f(x)\vert <\delta$$ . Then we have that $$\|k_{\varepsilon } {\ast} f - f\|_{L^{\infty }(K)} \rightarrow 0$$ as $$\varepsilon \rightarrow 0$$ . In particular, if f is bounded and continuous at a point x ₀ ∈ G, then $$(k_{\varepsilon } {\ast} f)(x_{0}) \rightarrow f(x_{0})$$ as $$\varepsilon \rightarrow 0$$ .

Proof.

We start with the case 1 ≤ p < ∞. We recall that continuous functions with compact support are dense in L p of locally compact Hausdorff spaces equipped with measures arising from nonnegative linear functionals; see [152, Theorem 12.10]. For a continuous function g supported in a compact set L we have we have $$\vert g(h^{-1}x) - g(x)\vert ^{p} \leq (2\|g\|_{L^{\infty }})^{p}\chi _{W^{-1}L}$$ for h in a relatively compact neighborhood W of the identity element e. By the Lebesgue dominated convergence theorem we obtain

(1.2.28)

as h → e. Now approximate a given f in L p (G) by a continuous function with compact support g to deduce that

(1.2.29)

Because of (1.2.29), given a δ > 0 there exists a neighborhood V of e such that

(1.2.30)

where c is the constant that appears in Definition 1.2.15 (i). Since $$k_{\varepsilon }$$ has integral one for all $$\varepsilon > 0$$, we have

(1.2.31)

Now take L p norms in x in (1.2.31). In view of (1.2.30),

(1.2.32)

while

(1.2.33)

provided we have that

(1.2.34)

Choose $$\varepsilon _{0} > 0$$ such that (1.2.34) is valid for $$\varepsilon <\varepsilon _{0}$$ by property (iii). Now (1.2.32) and (1.2.33) imply the required conclusion.

The case p = ∞ follows similarly. Let f be a bounded function on G that is uniformly continuous on K. Given $$\delta > 0$$, there is a neighborhood V of e such that, whenever y ∈ V and x ∈ K we have

(1.2.35)

where c is as in Definition 1.2.15 (i). By property (iii) in Definition 1.2.15, there is an $$\varepsilon _{0} > 0$$ such that for $$0 <\varepsilon <\varepsilon _{0}$$ we have

(1.2.36)

Using (1.2.35) and (1.2.36), we deduce that

This shows that $$k_{\varepsilon } {\ast} f$$ converge uniformly to f on K as $$\varepsilon \rightarrow 0$$. In particular, if K = { x ₀} and f is bounded and continuous at x ₀, we have $$(k_{\varepsilon } {\ast} f)(x_{0}) \rightarrow f(x_{0})$$. □ 

Remark 1.2.20.

Observe that if Haar measure satisfies (1.2.12), then the conclusion of Theorem 1.2.19 also holds for $$f {\ast} k_{\varepsilon }$$.

A simple modification in the proof of Theorem 1.2.19 yields the following variant, which presents a significant difference only when a = 0.

Theorem 1.2.21.

Let $$k_{\varepsilon }$$ be a family of functions on a locally compact group G that satisfies properties (i) and (iii) of Definition 1.2.15 and also

for some fixed $$a \in \mathbf{C}$$ and for all $$\varepsilon > 0$$ . Let f ∈ L p (G) for some 1 ≤ p ≤∞​.

(a)

If 1 ≤ p < ∞, then $$\|k_{\varepsilon } {\ast} f - af\|_{L^{p}(G)} \rightarrow 0$$ as $$\varepsilon \rightarrow 0\!$$ .

(b)

If $$p = \infty $$ and f is uniformly continuous on a subset K of G, in the sense that for any $$\delta > 0$$ there is a neighborhood V of the identity element of G such that $$\sup _{x\in G}\sup _{y\in V }\vert f(y^{-1}x) - f(x)\vert \leq \delta$$ , then we have that $$\|k_{\varepsilon } {\ast} f - af\|_{L^{\infty }(K)} \rightarrow 0$$ as $$\varepsilon \rightarrow 0$$ .

Exercises

1.2.1. Let G be a locally compact group and let f, g in L ¹(G) be supported in the subsets A and B of G, respectively. Prove that f ∗ g is supported in the algebraic product set AB.

1.2.2. For a function f on a locally compact group G and t ∈ G, let $$^{t}\!f(x) = f(tx)$$ and $$f^{t}(x) = f(xt)$$. Show that

whenever $$f,g \in L^{1}(G)$$, equipped with left Haar measure.

1.2.3. Let G be a locally compact group with left Haar measure. Let f ∈ L p (G) and $$\tilde{g} \in L^{p'}(G)$$, where 1 < p < ∞; recall that $$\widetilde{g}(x) = g(x^{-1})$$. For t, x ∈ G, let t g(x) = g(tx). Show that for any $$\varepsilon > 0$$ there exists a relatively compact symmetric neighborhood of the origin U such that u ∈ U implies $$\|^{u}\widetilde{g} -\widetilde{ g}\,\|_{L^{p'}(G)} <\varepsilon$$ and therefore

whenever v ⁻¹ w ∈ U.

1.2.4. (a) Prove that compactly supported functions are dense in $$L^{p}(\mathbf{R}^{n})$$ for all $$0 < p < \infty $$.

(b) Show that smooth functions with compact support are dense in $$L^{p}(\mathbf{R}^{n})$$ for all $$1 \leq p < \infty $$.

[Hint: Part (b): Use Theorem 1.2.19 with $$k_{\varepsilon }(x) =\varepsilon ^{-n}k(\varepsilon ^{-1}x)$$ and k smooth and compactly supported function.]

1.2.5. Show that a Haar measure λ for the multiplicative group of all positive real numbers is

1.2.6. Let $$G = \mathbf{R}^{2}\setminus \{(0,y):\,\, y \in \mathbf{R}\}$$ with group operation $$(x,y)(z,w) = (xz,xw + y)$$. [Think of G as the group of all 2 × 2 matrices with bottom row (0, 1) and nonzero top left entry.] Show that a left Haar measure on G is

while a right Haar measure on G is

1.2.7. ([144], [145]) Use Theorem 1.2.10 to prove that

when $$1 < p < \infty $$.

[Hint: On the multiplicative group $$(\mathbf{R}^{+}, \frac{dt} {t} )$$ consider the convolution of the function $$\vert f(x)\vert x^{\frac{1} {p} }$$ with the function $$x^{-\frac{1} {p'} }\chi _{[1,\infty )}$$ and the convolution of the function $$\vert f(x)\vert x^{1+ \frac{1} {p} }$$ with $$x^{\frac{1} {p} }\chi _{(0,1]}$$.]

1.2.8. (G. H. Hardy) Let $$0 < b < \infty $$ and $$1 \leq p < \infty $$. Prove that

[Hint: On the multiplicative group $$(\mathbf{R}^{+}, \frac{dt} {t} )$$ consider the convolution of the function $$\vert f(x)\vert x^{1-\frac{b} {p} }$$ with $$x^{-\frac{b} {p} }\chi _{[1,\infty )}$$ and of the function $$\vert f(x)\vert x^{1+ \frac{b} {p} }$$ with $$x^{\frac{b} {p} }\chi _{(0,1]}$$.]

1.2.9. On $$\mathbf{R}^{n}$$ let T(f) = f ∗ K, where K is a positive L ¹ function and f is in L p , $$1 \leq p \leq \infty $$. Prove that the operator norm of T: L p  → L p is equal to $$\|K\|_{L^{1}}$$.

[Hint: Clearly, $$\|T\|_{L^{p}\rightarrow L^{p}} \leq \| K\|_{L^{1}}$$. Conversely, fix $$0 <\varepsilon < 1$$ and let N be a positive integer. Let $$\chi _{N} =\chi _{B(0,N)}$$ and for any R > 0 let $$K_{R} = K\chi _{B(0,R)}$$, where B(x, R) is the ball of radius R centered at x. Observe that for $$\vert x\vert \leq (1-\varepsilon )N$$, we have $$B(0,N\varepsilon ) \subseteq B(x,N)$$; thus $$\int _{\mathbf{R}^{n}}\chi _{N}(x - y)K_{N\varepsilon }(y)\,dy =\int _{\mathbf{R}^{n}}K_{N\varepsilon }(y)\,dy =\big\| K_{N\varepsilon }\big\|_{L^{1}}$$. Then

Let $$N \rightarrow \infty $$ first and then $$\varepsilon \rightarrow 0$$.]

1.2.10. On the multiplicative group $$(\mathbf{R}^{+}, \frac{dt} {t} )$$ let T(f) = f ∗ K, where K is a positive L ¹ function and f is in L p , 1 ≤ p ≤ ∞. Prove that the operator norm of T: L p  → L p is equal to the L ¹ norm of K. Deduce that the constants p∕(p − 1) and p∕b are sharp in Exercises 1.2.7 and 1.2.8.

[Hint: Adapt the idea of Exercise 1.2.9 to this setting.]

1.2.11. Let $$Q_{k}(t) = c_{k}(1 - t^{2})^{k}$$ for t ∈ [−1, 1] and zero elsewhere, where c k is chosen such that $$\int _{-1}^{1}Q_{k}(t)\,dt = 1$$ for all $$k = 1,2,\ldots$$.

(a) Show that $$c_{k} < \sqrt{k}$$.

(b) Use part (a) to show that {Q k } k is an approximate identity on R as k → ∞.

(c) Given a continuous function f on R that vanishes outside the interval [−1, 1], show that f ∗ Q k converges to f uniformly on [−1, 1] as $$k \rightarrow \infty $$.

(d) (Weierstrass) Prove that every continuous function on [−1, 1] can be approximated uniformly by polynomials.

[Hint: Part (a): Estimate the integral $$\int _{ \vert t\vert \leq k^{-1/2}}Q_{k}(t)\,dt$$ from below using the inequality $$(1 - t^{2})^{k} \geq 1 - kt^{2}$$ for | t | ≤ 1. Part (d): Consider the function $$g(t) = f(t) - f(-1) -\frac{t+1} {2} (f(1) - f(-1)).$$]

1.2.12. Show that the Laplace transform $$L(f)(x) =\int _{ 0}^{\infty }f(t)e^{-xt}dt$$ maps $$L^{2}(0,\infty )$$ to itself with norm at most $$\sqrt{\pi }$$.

[Hint: Consider convolution with the kernel $$\sqrt{ t}\,e^{-t}$$ on the group $$L^{2}((0,\infty ), \frac{dt} {t} )$$.]

1.2.13. ([62]) Let $$F \geq 0$$, G ≥ 0 be measurable functions on the sphere $$\mathbf{S}^{n-1}$$ and let $$K \geq 0$$ be a measurable function on [−1, 1]. Prove that

where $$1 \leq p \leq \infty $$, $$\theta \cdot \varphi =\sum _{ j=1}^{n}\theta _{j}\varphi _{j}$$ and $$C =\int _{\mathbf{S}^{n-1}}K(\theta \cdot \varphi )\,d\varphi,$$ which is independent of θ. Moreover, show that C is the best possible constant in the preceding inequality. Using duality, compute the norm of the linear operator

from $$L^{p}(\mathbf{S}^{n-1})$$ to itself.

[Hint: Observe that $$\int _{\mathbf{S}^{n-1}}\int _{\mathbf{S}^{n-1}}F(\theta )G(\varphi )K(\theta \cdot \varphi )\,d\varphi \,d\theta$$ is bounded by the quantity

Apply Hölder’s inequality to the functions F and 1 with respect to the measure $$K(\theta \cdot \varphi )\,d\theta$$ to deduce that $$\int _{\mathbf{S}^{n-1}}F(\theta )K(\theta \cdot \varphi )\,d\theta$$ is controlled by

Use Fubini’s theorem to bound the latter by

Note that equality is attained if and only if both F and G are constants.]

1.3 Interpolation

The theory of interpolation of operators is vast and extensive. In this section we are mainly concerned with a couple of basic interpolation results that appear in a variety of applications and constitute the foundation of the field. These results are the Marcinkiewicz interpolation theorem and the Riesz–Thorin interpolation theorem. These theorems are traditionally proved using real and complex variables techniques, respectively. A byproduct of the Riesz–Thorin interpolation theorem, Stein’s theorem on interpolation of analytic families of operators, has also proved to be an important and useful tool in many applications and is presented at the end of the section.

We begin by setting up the background required to formulate the results of this section. Let (X, μ) and (Y, ν) be two measure spaces. Suppose we are given a linear operator T, initially defined on the set of simple functions on X, such that for all f simple on X, T(f) is a ν-measurable function on Y. Let $$0 < p < \infty $$ and $$0 < q < \infty $$. If there exists a constant C p, q  > 0 such that for all simple functions f on X we have

(1.3.1)

then by density, T admits a unique bounded extension from L p (X, μ) to L q (Y, ν). This extension is also denoted by T. Operators that map L p to L q are called of strong type (p, q) and operators that map L p to $$L^{q,\infty }$$ are called weak type (p, q) .

1.3.1 Real Method: The Marcinkiewicz Interpolation Theorem

Definition 1.3.1.

Let T be an operator defined on a linear space of complex-valued measurable functions on a measure space (X, μ) and taking values in the set of all complex-valued finite almost everywhere measurable functions on a measure space (Y, ν). Then T is called linear if for all f, g in the domain of T and all $$\lambda \in \mathbf{C}$$ we have

(1.3.2)

T is called sublinear if for all f, g in the domain of T and all $$\lambda \in \mathbf{C}$$ we have

(1.3.3)

T is called quasi-linear if for all f, g in the domain of T and all $$\lambda \in \mathbf{C}$$ we have

(1.3.4)

for some constant K > 0. Sublinearity is a special case of quasi-linearity.

For instance, T ₁ and T ₂ are linear operators, then $$(\vert T_{1}\vert ^{p} + \vert T_{2}\vert ^{p})^{1/p}$$ is sublinear if $$p \geq 1$$ and quasi-linear if 0 < p < 1.

Theorem 1.3.2.

Let (X,μ) be a $$\sigma$$ -finite measure space, let (Y,ν) be another measure space, and let $$0 < p_{0} < p_{1} \leq \infty $$ . Let T be a sublinear operator defined on $$L^{p_{0}}(X) + L^{p_{1}}(X) =\{ f_{0} + f_{1}:\,\, f_{j} \in L^{p_{j}}(X_{j}),j = 0,1\}$$ and taking values in the space of measurable functions on Y. Assume that there exist $$A_{0},A_{1} < \infty $$ such that

(1.3.5)

(1.3.6)

Then for all $$p_{0} < p < p_{1}$$ and for all f in $$L^{p}(X)$$ we have the estimate

(1.3.7)

where

(1.3.8)

Proof.

Assume first that p ₁ < ∞. Fix f a function in L p (X) and α > 0. We split $$f = f_{0}^{\alpha } + f_{1}^{\alpha }$$, where $$f_{0}^{\alpha }$$ is in $$L^{p_{0}}$$ and $$f_{1}^{\alpha }$$ is in $$L^{p_{1}}$$. The splitting is obtained by cutting | f | at height $$\delta \alpha$$ for some $$\delta > 0$$ to be determined later. Set

It can be checked easily that $$f_{0}^{\alpha }$$ (the unbounded part of f) is an $$L^{p_{0}}$$ function and that $$f_{1}^{\alpha }$$ (the bounded part of f) is an $$L^{p_{1}}$$ function. Indeed, since p ₀ < p, we have

and similarly, since p < p ₁,

In view of the subadditivity property of T contained in (1.3.3) we obtain that

which implies

and therefore

(1.3.9)

Hypotheses (1.3.5) and (1.3.6) together with (1.3.9) now give

In view of the last estimate and Proposition 1.1.4, we obtain that

and the convergence of the integrals in α is justified from p ₀ < p < p ₁, while the interchange of the integrals (Fubini’s theorem) uses the hypothesis that (X, μ) is a $$\sigma$$-finite measure space. We pick $$\delta > 0$$ such that

and observe that the last displayed constant is equal to the pth power of the constant in (1.3.8). We have therefore proved the theorem when $$p_{1} < \infty $$.

We now consider the case $$p_{1} = \infty $$. Write $$f = f_{0}^{\alpha } + f_{1}^{\alpha }$$, where

We have

provided we choose $$\gamma = (2A_{1})^{-1}$$. It follows that the set $$\{y \in Y \!:\, \vert T(f_{1}^{\alpha })(y)\vert >\alpha /2\}$$ has measure zero. Therefore,

Since T maps $$L^{p_{0}}$$ to $$L^{p_{0},\infty }$$ with norm at most A ₀, it follows that

(1.3.10)

Using (1.3.10) and Proposition 1.1.4, we obtain

This proves the theorem with constant

(1.3.11)

Observe that when $$p_{1} = \infty $$, the constant in (1.3.11) coincides with that in (1.3.8). □ 

Remark 1.3.3.

Notice that the proof of Theorem 1.3.2 only makes use of the subadditivity property $$\vert T(f\! +\! g)\vert \leq \vert T(f)\vert \! +\! \vert T(g)\vert $$ of T in hypothesis (1.3.3).

If T is a linear operator (instead of sublinear), then we can relax the hypotheses of Theorem 1.3.2 by assuming that (1.3.5) and (1.3.6) hold for all simple functions f on X. Then the functions $$f_{0}^{\alpha }$$ and $$f_{1}^{\alpha }$$ constructed in the proof are also simple, and we conclude that (1.3.7) holds for all simple functions f on X. By density, T has a unique extension on L p (X) that also satisfies (1.3.7).

1.3.2 Complex Method: The Riesz–Thorin Interpolation Theorem

The next interpolation theorem assumes stronger endpoint estimates, but yields a more natural bound on the norm of the operator on the intermediate spaces. Unfortunately, it is mostly applicable for linear operators and in some cases for sublinear operators (often via a linearization process) but it does not apply to quasi-linear operators without some loss in the constant.

Recall that a simple function is called finitely simple if it is supported in a set of finite measure. Finitely simple functions are dense in $$L^{p}(X,\mu )$$ for $$0 < p < \infty $$, whenever (X, μ) is a $$\sigma$$-finite measure space.

Theorem 1.3.4.

Let (X,μ) and (Y,ν) be two $$\sigma$$ -finite measure spaces. Let T be a linear operator defined on the set of all finitely simple functions on X and taking values in the set of measurable functions on Y. Let $$1 \leq p_{0},p_{1},q_{0},q_{1} \leq \infty $$ and assume that

(1.3.12)

for all finitely simple functions f on X. Then for all 0 < θ < 1 we have

(1.3.13)

for all finitely simple functions f on X, where

(1.3.14)

Consequently, when $$p < \infty $$ , by density, T has a unique bounded extension from L p (X,μ) to L q (Y,ν) when p and q are as in (1.3.14) .

Proof.

Let

be a finitely simple function on X, where a k  > 0, $$\alpha _{k}$$ are real, and A k are pairwise disjoint subsets of X with finite measure.

We need to control

where the supremum is taken over all finitely simple functions g on Y with L q′ norm less than or equal to 1. Write

where b j  > 0, $$\beta _{j}$$ are real, and B j are pairwise disjoint subsets of Y with finite ν-measure. Let

(1.3.15)

For z in the closed strip $$\overline{S} =\{ z \in \mathbf{C}:\, 0 \leq \mathrm{ Re}\,z \leq 1\}$$, define

(1.3.16)

and

Notice that f θ  = f and g θ  = f. By linearity we have

Since $$a_{k},b_{j} > 0$$, F is analytic in z, and the expression

is a finite constant, being an absolutely convergent integral; this is seen by Hölder’s inequality with exponents q ₀ and q ₀′ (or q ₁ and q ₁′) and (1.3.12).

By the disjointness of the sets A k we have (even when $$p_{0} = \infty $$)

since $$\vert a_{k}^{P(it)}\vert = a_{k}^{ \frac{p} {p_{0}} }$$, and by the disjointness of the B j ’s we have (even when q ₀ = 1)

since $$\vert b_{j}^{Q(it)}\vert = b_{j}^{ \frac{q'} {q_{0}'} }$$. Thus Hölder’s inequality and the hypothesis give

(1.3.17)

By similar calculations, which are valid even when $$p_{1} = \infty $$ and q ₁ = 1, we have

and

Also, in a way analogous to that we obtained (1.3.17) we deduce that

(1.3.18)

To finish the proof we will need the following lemma, known as Hadamard’s three lines lemma.

Lemma 1.3.5.

Let F be analytic in the open strip $$S =\{ z \in \mathbf{C}:\, 0 < \mbox{ Re}\ z < 1\}$$ , continuous and bounded on its closure, such that |F(z)|≤ B ₀ when Re z = 0 and |F(z)|≤ B ₁ when Re z = 1, for some $$0 < B_{0},B_{1} < \infty $$ . Then $$\vert F(z)\vert \leq B_{0}^{1-\theta }B_{1}^{\theta }$$ when Re z = θ, for any 0 ≤θ ≤ 1.

To prove the lemma we define analytic functions

for z in the unit strip S, for $$n = 1,2,\ldots$$. Since F is bounded on $$\overline{S}$$ and

for all $$z \in \overline{S}$$, we conclude that G is bounded by some constant M on $$\overline{S}$$. Since

we deduce that G n (x + iy) converges to zero uniformly in 0 ≤ x ≤ 1 as | y | → ∞. Select y(n) > 0 such that for | y | ≥ y(n), we have | G n (x + iy) | ≤ 1 for all x ∈ [0, 1]. Also, the assumptions on F imply that G is bounded by one on the two lines forming the boundary of $$\overline{S}$$. By the maximum principle we obtain that | G n (z) | ≤ 1 for all z in the rectangle [0, 1] × [−y(n), y(n)]; hence | G n (z) | ≤ 1 everywhere in the closed strip. Letting $$n \rightarrow \infty $$, we conclude that | G(z) | ≤ 1 in the closed strip. Taking z = θ + it we deduce that

whenever t is real. This proves the required conclusion. □ 

Returning to the proof of Theorem 1.3.4, we observe that F is analytic in the open strip S and continuous on its closure. Also, F is bounded on the closed unit strip (by some constant that depends on f and g). Therefore, (1.3.17), (1.3.18), and Lemma 1.3.5 give

when Re z = θ. Observe that P(θ) = Q(θ) = 1 and hence

Taking the supremum over all finitely simple functions g on Y with L q′ norm less than or equal to one, we conclude the proof of the theorem. □ 

We now give an application of Theorem 1.3.4.

Example 1.3.6.

One may prove Young’s inequality (Theorem 1.2.12) using the Riesz–Thorin interpolation theorem (Theorem 1.3.4). Fix a function g in L r and let T(f) = f ∗ g. Since $$T:\,\, L^{1} \rightarrow L^{r}$$ with norm at most $$\|g\|_{L^{r}}$$ and $$T:\,\, L^{r'} \rightarrow L^{\infty }$$ with norm at most $$\|g\|_{L^{r}}$$, Theorem 1.3.4 gives that T maps L p to L q with norm at most the quantity $$\|g\|_{L^{r}}^{\theta }\|g\|_{ L^{r}}^{1-\theta } =\| g\|_{ L^{r}}$$, where

(1.3.19)

Finally, observe that equations (1.3.19) give (1.2.13).

1.3.3 Interpolation of Analytic Families of Operators

Theorem 1.3.4 can be extended to the case in which the interpolated operators are allowed to vary. In particular, if a family of operators depends analytically on a parameter z, then the proof of this theorem can be adapted to work in this setting.

We describe the setup for this theorem. Let (X, μ) and (Y, ν) be $$\sigma$$-finite measure spaces. Suppose that for every z in the closed strip $$\overline{S} =\{ z \in \mathbf{C}:\, 0 \leq \mathrm{ Re}\,z \leq 1\}$$ there is an associated linear operator T z defined on the space of finitely simple functions on X and taking values in the space of measurable functions on Y such that

(1.3.20)

whenever A and B are subsets of finite measure of X and Y, respectively. The family {T z } z is said to be analytic if for all f, g finitely simple functions we have that the function

(1.3.21)

is analytic in the open strip $$S =\{ z \in \mathbf{C}:\, 0 <\mathrm{ Re}\,z < 1\}$$ and continuous on its closure. The analytic family {T z } z is called of admissible growth if there is a constant τ ₀ with $$0 \leq \tau _{0} <\pi$$ such that for finitely simple functions f on X and g on Y there is constant C(f, g) such that

(1.3.22)

for all z satisfying 0 ≤ Re z ≤ 1. Note that if there is τ ₀ ∈ (0, π) such that for all measurable subsets A of X and B of Y of finite measure there is a constant c(A, B) such that

(1.3.23)

then (1.3.22) holds for $$f =\sum _{ k=1}^{M}a_{k}\chi _{A_{k}}$$ and $$g =\sum _{ j=1}^{N}b_{j}\chi _{B_{j}}$$ and

The extension of the Riesz–Thorin interpolation theorem is as follows.

Theorem 1.3.7.

Let T z be an analytic family of linear operators of admissible growth defined on the space of finitely simple functions of a $$\sigma$$ -finite measure space (X,μ) and taking values in the set of measurable functions of another $$\sigma$$ -finite measure space (Y,ν). Let $$1 \leq p_{0},p_{1},q_{0},q_{1} \leq \infty $$ and suppose that M ₀ and M ₁ are positive functions on the real line such that for some τ ₁ with $$0 \leq \tau _{1} <\pi$$ we have

(1.3.24)

for j = 0,1. Fix 0 < θ < 1 and define p,q by the equations

(1.3.25)

Suppose that for all finitely simple functions f on X we have

(1.3.26)

(1.3.27)

Then for all finitely simple functions f on X we have

(1.3.28)

where for 0 < x < 1

Thus, by density, T θ has a unique bounded extension from L p (X,μ) to L q (Y,ν) when p and q are as in (1.3.25) .

Note that in view of (1.3.24), the integral defining M(t) converges absolutely. The proof of the previous theorem is based on an extension of Lemma 1.3.5.

Lemma 1.3.8.

Let F be analytic on the open strip $$S =\{ z \in \mathbf{C}:\, 0 < \mbox{ Re}\,\,z < 1\}$$ and continuous on its closure such that for some $$A < \infty $$ and $$0 \leq \tau _{0} <\pi$$ we have

(1.3.29)

for all $$z \in \overline{S}$$ . Then

whenever 0 < x < 1, and y is real.

Assuming Lemma 1.3.8, we prove Theorem 1.3.7.

Proof.

Fix 0 < θ < 1 and finitely simple functions f on X and g on Y such that $$\|f\|_{L^{p}} =\| g\|_{L^{q'}} = 1$$. Note that since 0 < θ < 1 we must have $$1 < p,q < \infty $$. Let

where a k  > 0, b j  > 0, $$\alpha _{k}$$, $$\beta _{j}$$ are real, A k are pairwise disjoint subsets of X with finite measure, and B j are pairwise disjoint subsets of Y with finite measure for all k, j. Let P(z), Q(z) be as in (1.3.15) and f z , g z as in (1.3.16). Define for $$z \in \overline{S}$$

(1.3.30)

Linearity gives that

and conditions (1.3.20) together with the fact that $$\{T_{z}\}_{z}$$ is an analytic family imply that F(z) is a well-defined analytic function on the unit strip that extends continuously to its boundary.

Since $$\{T_{z}\}_{z}$$ is a family of admissible growth, (1.3.23) holds for some c(A k , B j ) and $$\tau _{0} \in (0,\pi )$$ and this combined with the facts that

for all z with $$0 < \mbox{ Re}\,\,z < 1$$, implies (1.3.29) with τ ₀ as in (1.3.23) and

Thus F satisfies the hypotheses of Lemma 1.3.8. Moreover, the calculations in the proof of Theorem 1.3.4 show that (even when p ₀ = ∞, q ₀ = 1, $$p_{1} = \infty $$, q ₁ = 1)

(1.3.31)

(1.3.32)

Hölder’s inequality, (1.3.31), and the hypothesis (1.3.26) now give

for all y real. Similarly, (1.3.32), and (1.3.27) imply

for all $$y \in \mathbf{R}$$. These inequalities and the conclusion of Lemma 1.3.8 yield

for all 0 < x < 1. But notice that

(1.3.33)

Taking absolute values and the supremum over all finitely simple functions g on Y with L q′ norm equal to one, we conclude the proof of (1.3.28) for finitely simple functions f with L p norm one. Then (1.3.28) follows by replacing f by $$f/\|f\|_{L^{p}}$$. □ 

We end this section with the proof of Lemma 1.3.8.

Proof of Lemma 1.3.8. Recall the Poisson integral formula

(1.3.34)

which is valid for a harmonic function U defined on the unit disk D = { z:   | z |  < 1} when | z |  < R < 1. See [307, p. 258].

Consider now a subharmonic function u on D that is continuous on the circle $$\vert \zeta \vert = R < 1$$. When U = u, the right side of (1.3.34) defines a harmonic function on the set $$\{z \in \mathbf{C}:\,\, \vert z\vert < R\}$$ that coincides with u on the circle $$\vert \zeta \vert = R$$. The maximum principle for subharmonic functions ([307, p. 362]) implies that for | z |  < R < 1 we have

(1.3.35)

This is valid for all subharmonic functions u on D that are continuous on the circle $$\vert \zeta \vert = R$$ when ρ < R < 1.

It is not difficult to verify that

is a conformal map from D onto the strip $$S = (0,1) \times \mathbf{R}$$. Indeed, $$i(1+\zeta )/(1-\zeta )$$ lies in the upper half-plane and the preceding complex logarithm is a well defined holomorphic function that takes the upper half-plane onto the strip $$\mathbf{R} \times (0,\pi )$$. Since $$F \circ h$$ is a holomorphic function on D, log | F ∘ h | is a subharmonic function on D. Applying (1.3.35) to the function z↦log | F(h(z)) | , we obtain

(1.3.36)

when z = ρ e i θ and | z |  = ρ < R. Observe that when $$\vert \zeta \vert = 1$$ and $$\zeta \neq \pm 1$$, $$h(\zeta )$$ has real part zero or one. It follows from the hypothesis that

Therefore, $$\log \vert F(h(\zeta ))\vert $$ is bounded by a multiple of $$\vert 1 +\zeta \vert ^{-\tau _{0}/\pi }\vert 1 -\zeta \vert ^{-\tau _{0}/\pi }$$, which is integrable over the set $$\vert \zeta \vert = 1$$, since $$\tau _{0} <\pi$$. Fix now $$z =\rho e^{i\theta }$$ with ρ < R and let R → 1 in (1.3.36). The Lebesgue dominated convergence theorem gives that

(1.3.37)

Setting x = h(ρ e i θ ), we obtain that

from which it follows that $$\rho = (\cos (\pi x))/(1 +\sin (\pi x))$$ and θ = −π∕2 when $$0 < x \leq \frac{1} {2}$$, while $$\rho = -(\cos (\pi x))/(1 +\sin (\pi x))$$ and θ = π∕2 when $$\frac{1} {2} \leq x < 1$$. In either case we easily deduce that

Using this we write (1.3.37) as

(1.3.38)

We now change variables. On the interval [−π, 0) we use the change of variables $$it = h(e^{i\varphi })$$ or, equivalently, $$e^{i\varphi } = -\tanh (\pi t) - i\,\mbox{ sech}\,\!(\pi t)$$. Observe that as $$\varphi$$ ranges from −π to 0, t ranges from $$+\infty $$ to $$-\infty $$. Furthermore, $$d\varphi = -\pi \,\mbox{ sech}\,\!(\pi t)\,dt$$. We have

(1.3.39)

On the interval (0, π] we use the change of variables $$1 + it = h(e^{i\varphi })$$ or, equivalently, $$e^{i\varphi } = -\tanh (\pi t) + i\,\mbox{ sech}\,\!(\pi t)$$. Observe that as $$\varphi$$ ranges from 0 to π, t ranges from $$-\infty $$ to $$+\infty $$. Furthermore, $$d\varphi =\pi \, \mbox{ sech}\,\!(\pi t)\,dt$$. Similarly, we obtain

(1.3.40)

Adding (1.3.39) and (1.3.40) and using (1.3.38) we conclude the proof when y = 0.

We now consider the case where y ≠ 0. Fix y ≠ 0 and define the function G(z) = F(z + iy). Then G is analytic on the open strip $$S =\{ z \in \mathbf{C}:\, 0 < \mbox{ Re}\,\,z < 1\}$$ and continuous on its closure. Moreover, for some $$A < \infty $$ and 0 ≤ τ ₀ < π we have

for all $$z \in \overline{S}$$. Then the case y = 0 for G (with A replaced by $$Ae^{\tau _{0}\vert y\vert }$$) yields

which yields the required conclusion for any real y, since G(x) = F(x + iy), G(it) = F(it + iy), and G(1 + it) = F(1 + it + iy). □ 

Exercises

1.3.1. Generalize Theorem 1.3.2 to the situation in which T is quasi-subadditive, that is, it satisfies for some K > 0,

for all f, g in the domain of T. Prove that in this case, the constant A in (1.3.7) can be taken to be K times the constant in (1.3.8).

1.3.2. Let (X, μ), (Y, ν) be two $$\sigma$$-finite measure spaces. Let $$1 < p < r \leq \infty $$ and suppose that T be a sublinear operator defined on the space $$L^{p_{0}}(X) + L^{p_{1}}(X)$$ and taking values in the space of measurable functions on Y. Assume that T maps L ¹(X) to $$L^{1,\infty }(Y )$$ with norm A ₀ and L r (X) to L r (Y ) with norm A ₁. Let $$0 < p_{0} < p_{1} \leq \infty $$. Prove that T maps $$L^{p}$$ to $$L^{p}$$ with norm at most

[Hint: First interpolate between L ¹ and L r using Theorem 1.3.2 and then interpolate between $$L^{\frac{p+1} {2} }$$ and L r using Theorem 1.3.4.]

1.3.3. Let $$0 < p_{0} < p < p_{1} \leq \infty $$ and let T be an operator as in Theorem 1.3.2 that also satisfies

for all $$f \in L^{p_{0}} + L^{p_{1}}$$.

(a) If p ₀ = 1 and $$p_{1} = \infty $$, prove that T maps L p to L p with norm at most

(b) More generally, if $$p_{0} < p < \infty $$, prove that the norm of T from L p to L p is at most

where $$B(s,t) =\int _{ 0}^{1}x^{s-1}(1 - x)^{t-1}\,dx$$ is the usual Beta function.

(c) When $$0 < p_{0} < p_{1} < \infty $$, then the norm of T from L p to L p is at most

[Hint: The hypothesis | T(f) | ≤ T( | f | ) reduces matters to nonnegative functions. Parts (a), (b): Given $$f \geq 0$$ and $$\alpha > 0$$ write f = f ₀ + f ₁, where $$f_{0} = f -\lambda \alpha /A_{1}$$ when $$f \geq \lambda \alpha /A_{1}$$ and zero otherwise. Here $$0 <\lambda < 1$$ to be chosen later. Then we have that $$\vert \{\vert T(f)\vert >\alpha \} \vert \leq \vert \{\vert T(f_{0})\vert > (1-\lambda )\alpha \}\vert $$. Part (c): Write $$f = f_{0} + f_{1}$$, where $$f_{0} = f-\delta \alpha$$ when $$f \geq \delta \alpha$$ and zero otherwise. Use that

and optimize over $$\delta > 0$$.]

1.3.4. Let $$0 \leq \gamma,\delta <\pi$$. For every $$z \in S_{a,b} =\{ z \in \mathbf{C}:\, a <\mathrm{ Re}\,z < b\}$$, let T z be a family of linear operators defined on finetely simple functions on a $$\sigma$$-finite measure space (X, μ) and taking values in another $$\sigma$$-finite measure space (Y, ν). Assume that $$\{T_{z}\}_{z}$$ is an analytic on of S a, b , in the sense of (1.3.21), continuous on its closure, and that for all simple functions f on X and g on Y there is a constant $$C_{f,g} < \infty $$ such that for all z ∈ S a, b ,

Let $$1 \leq p_{0},q_{0},p_{1},q_{1} \leq \infty $$. Suppose that T a+iy maps $$L^{p_{0}}(X)$$ to $$L^{q_{0}}(Y )$$ with bound M ₀(y) and T b+iy maps $$L^{p_{1}}(X)$$ to $$L^{q_{1}}(Y )$$ with bound M ₁(y), where

Then for a < t < b, T t maps L p (X) to L q (Y ), where

1.3.5. ([331]) On $$\mathbf{R}^{n}$$ let $$x = (x_{1},\ldots,x_{n})$$ and $$\vert x\vert = (x_{1}^{2} +\ldots +x_{n}^{2})^{1/2}$$. Let

where $$\lambda$$ is a complex number and $$J_{\lambda +\frac{n} {2} }$$ is the Bessel function of order $$\lambda +\frac{n} {2}$$. Let $$T_{\lambda }$$ be the operator given by convolution with $$K_{\lambda }$$. Show that $$T_{\lambda }$$ maps $$L^{p}(\mathbf{R}^{n})$$ to itself for $$\mathrm{Re}\,\lambda > (n - 1)\vert \frac{1} {2} -\frac{1} {p}\vert $$.

[Hint: In view of the calculation of the Fourier transform of $$K_{\lambda }$$ contained in Appendix B.5, we have that when $$\mathrm{Re}\,\lambda = 0$$, $$T_{\lambda }$$ maps $$L^{2}(\mathbf{R}^{n})$$ to itself with norm 1. Using the estimates in Appendices B.6 and B.7, conclude that $$K_{\lambda }$$ is integrable and thus $$T_{\lambda }$$ maps $$L^{1}(\mathbf{R}^{n})$$ to itself with an appropriate constant when $$\mbox{ Re}\,\,\lambda = (n - 1)/2+\delta$$ (for $$\delta > 0$$). Then use Exercise 1.3.4.]

1.3.6. Observe that Theorem 1.3.7 yields the stronger conclusion

for $$z \in S =\{ z \in \mathbf{C}:\, 0 < \mbox{ Re}\,\,z < 1\}$$, where for z = x + iy

1.3.7. ([380]) Let (X, μ) and (Y, ν) be two measure spaces with $$\mu (X) < \infty $$ and $$\nu (Y ) < \infty $$. Let T be a countably subadditive operator that maps L p (X) to L p (Y ) for every 1 < p ≤ 2 with norm $$\|T\|_{L^{p}\rightarrow L^{p}} \leq A(p - 1)^{-\alpha }$$ for some fixed $$A,\alpha > 0$$. (Countably subadditive means that $$\vert T(\sum _{j}f_{j})\vert \leq \sum _{j}\vert T(f_{j})\vert $$ for all f j in L p (X) with $$\sum _{j}f_{j} \in L^{p}$$.) Prove that for all f measurable on X we have

where $$C_{\alpha } =\sum _{ k=1}^{\infty }k^{\alpha }(2/3)^{k}$$. This result provides an example of extrapolation.

[Hint: Write

where $$S_{k} =\{ 2^{k} \leq \vert f\vert < 2^{k+1}\}$$ when $$k \geq 1$$ and S ₀ = { | f |  < 2}. Using Hölder’s inequality and the hypotheses on T, obtain that

for k ≥ 1. Note that for $$k \geq 1$$ we have $$\nu (Y )^{ \frac{1} {k+1} } \leq \max (1,\nu (Y ))^{\frac{1} {2} }$$ and consider the cases $$\mu (S_{k}) \geq 3^{-k-1}$$ and μ(S k ) ≤ 3−k−1 when summing in k ≥ 1. The term with k = 0 is easier.]

1.3.8. Prove that for 0 < x < 1 we have

and conclude that Lemma 1.3.8 reduces to Lemma 1.3.5 when the functions M ₀(y) and M ₁(y) are constant and assumption (1.3.29) is replaced by the stronger assumption that F is bounded on $$\overline{S}$$.

[Hint: In the first integral write $$\cosh (\pi t) = \frac{1} {2}(e^{\pi t} + e^{-\pi t})$$. Then use the change of variables $$s = e^{\pi t}$$.]

1.3.9. Let (X, μ), (Y, ν) be $$\sigma$$-finite measure spaces, and let $$0 < p_{0} < p_{1} \leq \infty $$. Let T be a sublinear operator defined on the space $$L^{p_{0}}(X) + L^{p_{1}}(X)$$ and taking values in the space of measurable functions on Y. Suppose T is a sublinear operator such that maps $$L^{p_{0}}$$ to L ∞ with constant A ₀ and $$L^{p_{1}}$$ to L ∞ with constant A ₁. Prove T maps L p to L ∞ with constant $$2A_{0}^{1-\theta }A_{1}^{\theta }$$ where

1.4 Lorentz Spaces

Suppose that f is a measurable function on a measure space (X, μ). It would be desirable to have another function f ∗ defined on [0, ∞) that is decreasing and equidistributed with f. By this we mean

(1.4.1)

for all $$\alpha \geq 0$$. This is achieved via a simple construction discussed in this section.

1.4.1 Decreasing Rearrangements

Definition 1.4.1.

Let f be a complex-valued function defined on X. The decreasing rearrangement of f is the function f ∗ defined on [0, ∞) by

(1.4.2)

We adopt the convention $$\inf \varnothing = \infty $$, thus having f ∗(t) = ∞ whenever d f (α) > t for all $$\alpha \geq 0$$. Observe that f ∗ is decreasing and supported in [0, μ(X)].

Before we proceed with properties of the function f ∗, we work out three examples.

Fig. 1.3

The graph of a simple function f(x) and its decreasing rearrangement f ∗(t).

Example 1.4.2.

Consider the simple function of Example 1.1.2,

where E j are pairwise disjoint sets of finite measure and $$a_{1} >\ldots > a_{N} > 0$$. We saw in Example 1.1.2 that

where

and $$a_{N+1} = B_{0} = 0$$ and $$a_{0} = \infty $$. Observe that for $$B_{0} \leq t < B_{1}$$, the smallest s > 0 with d f (s) ≤ t is a ₁. Similarly, for B ₁ ≤ t < B ₂, the smallest s > 0 with d f (s) ≤ t is a ₂. Arguing this way, it is not difficult to see that

See Figure 1.3.

Example 1.4.3.

On $$(\mathbf{R}^{n},dx)$$ let

A computation shows that

and therefore

where v n is the volume of the unit ball in $$\mathbf{R}^{n}$$.

Example 1.4.4.

Again on $$(\mathbf{R}^{n},dx)$$ let $$g(x) = 1 - e^{-\vert x\vert ^{2} }$$. We can easily see that $$d_{g}(\alpha ) = 0$$ if α ≥ 1 and d g (α) = ∞ if α < 1. We conclude that g ∗(t) = 1 for all t ≥ 0. This example indicates that although quantitative information is preserved, significant qualitative information is lost in passing from a function to its decreasing rearrangement.

It is clear from the previous examples that f ∗ is continuous from the right and decreasing. The following are some properties of the function f ∗.

Proposition 1.4.5.

For f, g, f n μ-measurable, $$k \in \mathbf{C}$$ , and $$0 \leq t,s,t_{1},t_{2} < \infty $$ we have

(1)

$$\quad f^{{\ast}}(d_{f}(\alpha )) \leq \alpha$$ whenever α > 0.

(2)

$$\quad d_{f}(f^{{\ast}}(t)) \leq t$$ .

(3)

$$\quad f^{{\ast}}(t) > s$$ if and only if $$t < d_{f}(s)$$ ; that is, $$\{t \geq 0:\,\, f^{{\ast}}(t) > s\} = [0,d_{f}(s))$$ .

(4)

$$\quad \vert g\vert \leq \vert f\vert $$ μ-a.e. implies that $$g^{{\ast}}\leq f^{{\ast}}$$ and |f| ∗ = f ∗ .

(5)

$$\quad (kf)^{{\ast}} = \vert k\vert f^{{\ast}}$$ .

(6)

$$\quad (f + g)^{{\ast}}(t_{1} + t_{2}) \leq f^{{\ast}}(t_{1}) + g^{{\ast}}(t_{2})$$ .

(7)

$$\quad (fg)^{{\ast}}(t_{1} + t_{2}) \leq f^{{\ast}}(t_{1})g^{{\ast}}(t_{2})$$ .

(8)

$$\quad \vert f_{n}\vert \uparrow \vert f\vert $$ μ-a.e. implies $$f_{n}^{{\ast}}\uparrow f^{{\ast}}$$ .

(9)

$$\quad \vert f\vert \leq \liminf \limits _{n\rightarrow \infty }\vert f_{n}\vert $$ μ-a.e. implies $$f^{{\ast}}\leq \liminf \limits _{n\rightarrow \infty }f_{n}^{{\ast}}$$ .

(10)

$$\quad f^{{\ast}}$$ is right continuous on [0,∞).

(11)

If $$f^{{\ast}}(t) < \infty $$ , c > 0, and $$\mu (\{\vert f\vert \geq f^{{\ast}}(t)\! -\! c\}) < \infty $$ , then $$t \leq \mu (\{\vert f\vert \geq f^{{\ast}}(t)\})$$ .

(12)

$$\quad d_{f} = d_{f^{{\ast}}}$$ .

(13)

$$\quad (\vert f\vert ^{p})^{{\ast}} = (f^{{\ast}})^{p}$$ when 0 < p < ∞​.

(14)

$$\quad \int _{X}\vert f\vert ^{p}\,d\mu =\int _{ 0}^{\infty }f^{{\ast}}(t)^{p}\,dt$$ when 0 < p < ∞​.

(15)

$$\quad \big\|f\big\|_{L^{\infty }} = f^{{\ast}}(0)$$ .

(16)

$$\quad \sup \limits _{t>0}t^{q}f^{{\ast}}(t) =\sup \limits _{\alpha >0}\alpha \left (d_{f}(\alpha )\right )^{q}$$ for 0 < q < ∞​.

Proof.

Property (1): The set $$A =\{ s > 0:\,\, d_{f}(s) \leq d_{f}(\alpha )\}$$ contains $$\alpha$$ and thus $$f^{{\ast}}(d_{f}(\alpha )) =\inf A \leq \alpha$$.

Property (2): Let $$s_{n} \in \{ s > 0:\, d_{f}(s) \leq t\}$$ be such that $$s_{n} \downarrow f^{{\ast}}(t)$$. Then $$d_{f}(s_{n}) \leq t$$, and the right continuity of d f (Exercise 1.1.1 (a)) implies that $$d_{f}(f^{{\ast}}(t)) \leq t$$.

Property (3): If $$s < f^{{\ast}}(t) =\inf \{ u > 0:\,\, d_{f}(u) \leq t\}$$, then $$s\notin \{u > 0:\,\, d_{f}(u) \leq t\}$$ which gives d f (s) > t. Conversely, if for some t < d f (s) we had f ∗(t) ≤ s, applying d f and using property (2) would yield the contradiction $$d_{f}(s) \leq d_{f}(f^{{\ast}}(t)) \leq t$$.

Properties (4) and (5) are left to the reader.

Properties (6) and (7): Let $$A =\{ s_{1} > 0:\,\, d_{f}(s_{1}) \leq t_{1}\}$$, $$B =\{ s_{2} > 0:\,\, d_{g}(s_{2}) \leq t_{2}\}$$, $$P =\{ s > 0:\,\, d_{fg}(s) \leq t_{1} + t_{2}\}$$, and $$S =\{ s > 0:\,\, d_{f+g}(s) \leq t_{1} + t_{2}\}$$. Then $$A + B \subseteq S$$ and $$A \cdot B \subseteq P$$; thus $$(f + g)^{{\ast}}(t_{1} + t_{2}) =\inf S \leq s_{1} + s_{2}$$ and $$(fg)^{{\ast}}(t_{1} + t_{2}) =\inf P \leq s_{1}s_{2}$$ are valid for all s ₁ ∈ A and s ₂ ∈ B. Taking the infimum over all s ₁ ∈ A and s ₂ ∈ B yields the conclusions.

Property (8): It follows from the definition of decreasing rearrangements that $$f_{n}^{{\ast}}\leq f_{n+1}^{{\ast}}\leq f^{{\ast}}$$ for all n. Let $$h =\lim _{n\rightarrow \infty }f_{n}^{{\ast}}$$; then obviously h ≤ f ∗. Since $$f_{n}^{{\ast}}\leq h$$, we have $$d_{f_{n}}(h(t)) \leq d_{f_{n}}(f_{n}^{{\ast}}(t)) \leq t$$, which implies, in view of Exercise 1.1.1 (c), that d f (h(t)) ≤ t by letting $$n \rightarrow \infty $$. It follows that f ∗ ≤ h, hence $$h = f^{{\ast}}$$.

Property (9): Set $$F_{n} =\inf _{m\geq n}\vert f_{m}\vert $$ and $$h =\liminf _{n\rightarrow \infty }\vert f_{n}\vert =\sup _{n\geq 1}F_{n}$$. Since F n ↑ h, property (8) yields that $$F_{n}^{{\ast}}\uparrow h^{{\ast}}$$ as $$n \rightarrow \infty $$. By hypothesis we have | f | ≤ h, hence $$f^{{\ast}}\leq h^{{\ast}} =\sup _{n}F_{n}^{{\ast}}$$. Since F n  ≤ | f m  | for m ≥ n, it follows that $$F_{n}^{{\ast}}\leq f_{m}^{{\ast}}$$ for m ≥ n; thus $$F_{n}^{{\ast}}\leq \inf _{m\geq n}f_{m}^{{\ast}}$$. Putting these facts together, we obtain $$f^{{\ast}}\leq h^{{\ast}}\leq \sup _{n}\inf _{m\geq n}f_{m}^{{\ast}} =\liminf _{n\rightarrow \infty }f_{n}^{{\ast}}$$.

Property (10): If f ∗(t ₀) = 0, then f ∗(t) = 0 for all t > t ₀ and thus f ∗ is right continuous at t ₀. Suppose f ∗(t ₀) > 0. Pick $$\alpha$$ such that $$0 <\alpha < f^{{\ast}}(t_{0})$$ and let $$\{t_{n}\}_{n=1}^{\infty }$$ be a sequence of real numbers decreasing to zero. The definition of f ∗ yields that $$d_{f}(f^{{\ast}}(t_{0})-\alpha ) > t_{0}$$. Since t n ↓ 0, there is an $$n_{0} \in \mathbf{Z}^{+}$$ such that $$d_{f}(f^{{\ast}}(t_{0})-\alpha ) > t_{0} + t_{n}$$ for all n ≥ n ₀. Property (3) yields that for all n ≥ n ₀ we have $$f^{{\ast}}(t_{0})-\alpha < f^{{\ast}}(t_{0} + t_{n})$$, and since the latter is at most f ∗(t ₀), the right continuity of f ∗ follows.

Property (11): The definition of f ∗ yields that the set $$A_{n} =\{ \vert f\vert > f^{{\ast}}(t) - c/n\}$$ has measure μ(A n ) > t. The sets A n form a decreasing sequence as n increases and $$\mu (A_{1}) < \infty $$ by assumption. Consequently, $$\{\vert f\vert \geq f^{{\ast}}(t)\} =\bigcap _{ n=1}^{\infty }A_{n}$$ has measure greater than or equal to t.

Property (12): This is immediate for nonnegative simple functions in view of Examples 1.1.2 and 1.4.2. For an arbitrary measurable function f, find a sequence of nonnegative simple functions f n such that f n ↑ | f | and apply (9).

Property (13): It follows from $$d_{\vert f\vert ^{p}}(\alpha ) = d_{f}(\alpha ^{1/p}) = d_{f^{{\ast}}}(\alpha ^{1/p}) = d_{(f^{{\ast}})^{p}}(\alpha )$$ for all $$\alpha > 0$$.

Property (14): This is a consequence of property (12) and of Proposition 1.1.4.

Property (15): This is a restatement of (1.1.2).

Property (16): Given $$\alpha > 0$$, without loss of generality we may assume $$d_{f}(\alpha ) > 0$$. Pick $$\varepsilon$$ satisfying $$0 <\varepsilon < d_{f}(\alpha )$$. Property (3) yields $$f^{{\ast}}(d_{f}(\alpha )-\varepsilon ) >\alpha$$, which implies that

We first let $$\varepsilon \rightarrow 0$$ and then take the supremum over all $$\alpha > 0$$ to obtain one direction. Conversely, given t > 0, assume without loss of generality that f ∗(t) > 0, and pick $$\varepsilon$$ such that $$0 <\varepsilon < f^{{\ast}}(t)$$. Property (3) yields $$d_{f}(f^{{\ast}}(t)-\varepsilon ) > t$$. This implies that $$\sup _{\alpha >0}\alpha (d_{f}(\alpha ))^{q} \geq (f^{{\ast}}(t)-\varepsilon )(d_{f}(f^{{\ast}}(t)-\varepsilon ))^{q} > (f^{{\ast}}(t)-\varepsilon )t^{q}$$. We first let $$\varepsilon \rightarrow 0$$ and then take the supremum over all t > 0 to obtain the opposite direction of the claimed equality. □ 

1.4.2 Lorentz Spaces

Having disposed of the basic properties of decreasing rearrangements of functions, we proceed with the definition of the Lorentz spaces.

Definition 1.4.6.

Given f a measurable function on a measure space (X, μ) and 0 < p, q ≤ ∞, define

The set of all f with $$\|f\|_{L^{p,q}} < \infty $$ is denoted by L p, q (X, μ) and is called the Lorentz space with indices p and q.

As in L p and in weak L p , two functions in L p, q (X, μ) are considered equal if they are equal μ-almost everywhere. Observe that the previous definition implies that $$L^{\infty,\infty } = L^{\infty }$$, L p, ∞  = weak L p in view of Proposition 1.4.5 (16) and that $$L^{p,p} = L^{p}$$.

Remark 1.4.7.

Observe that for all $$0 < p,r < \infty $$ and $$0 < q \leq \infty $$ we have

(1.4.3)

On $$\mathbf{R}^{n}$$ let $$\delta ^{\varepsilon }(f)(x) = f(\varepsilon x)$$, $$\varepsilon > 0$$, be the dilation operator. It is straightforward that $$d_{\delta ^{\varepsilon }(f)}(\alpha ) =\varepsilon ^{-n}d_{f}(\alpha )$$ and $$(\delta ^{\varepsilon }(f))^{{\ast}}(t) = f^{{\ast}}(\varepsilon ^{n}t)$$. It follows that Lorentz norms satisfy the following dilation identity:

(1.4.4)

Next, we calculate the Lorentz norms of a finitely simple function.

Example 1.4.8.

Using the notation of Example 1.4.2, when $$0 < p,q < \infty $$ we have

and also

Next, we calculate $$\|f\|_{L^{\infty,q}}$$ for the simple function f of Example 1.4.2. when $$q < \infty $$. It turns out that

since B ₀ = 0. We conclude that the only nonnegative simple function with finite L ∞, q norm is the zero function. Given a general nonzero function g ∈ L ∞, q with 0 < q < ∞, there is a nonzero simple function s with 0 ≤ s ≤ g. Then s has infinite norm, and therefore so does g. We deduce that L ∞, q (X) = { 0} when 0 < q < ∞.

Proposition 1.4.9.

For $$0 < p < \infty $$ and $$0 < q \leq \infty $$ , we have the identity

(1.4.5)

Proof.

The case $$q = \infty $$ is statement (16) in Proposition 1.4.5, and we may therefore concentrate on the case $$q < \infty $$. If f is the simple function of Example 1.1.2, then

with the understanding that a N+1 = 0. Using the this formula and identity in Example 1.4.8, we obtain the validity of (1.4.5) for simple functions. In general, given a measurable function f, find a sequence of nonnegative simple functions such that f n ↑ | f | a.e. Then $$d_{f_{n}} \uparrow d_{f}$$ (Exercise 1.1.1 (c)) and $$f_{n}^{{\ast}}\uparrow f^{{\ast}}$$ (Proposition 1.4.5 (8)). Using the Lebesgue monotone convergence theorem we deduce (1.4.5). □ 

Since $$L^{p,p} \subseteq L^{p,\infty }$$, one may wonder whether these spaces are nested. The next result shows that for any fixed p, the Lorentz spaces L p, q increase as the exponent q increases.

Proposition 1.4.10.

Suppose 0 < p ≤∞ and 0 < q < r ≤∞. Then there exists a constant c p,q,r (which depends on p, q, and r) such that

(1.4.6)

In other words, L p,q is a subspace of L p,r .

Proof.

We may assume $$p < \infty $$, since the case $$p = \infty $$ is trivial. We have

Hence, taking the supremum over all t > 0, we obtain

(1.4.7)

This establishes (1.4.6) in the case r = ∞. Finally, when r < ∞, we have

(1.4.8)

Inequality (1.4.7) combined with (1.4.8) gives (1.4.6) with $$c_{p,q,r} = (q/p)^{(r-q)/rq}$$. □ 

Unfortunately, the functionals $$\|\,\cdot \,\|_{L^{p,q}}$$ do not satisfy the triangle inequality. For instance, consider the functions f(t) = t and g(t) = 1 − t defined on [0, 1]. Then $$f^{{\ast}}(\alpha ) = g^{{\ast}}(\alpha ) = (1-\alpha )\chi _{[0,1]}(\alpha )$$. A simple calculation shows that the inequality $$\|f + g\|_{L^{p,q}} \leq \| f\|_{L^{p,q}} +\| g\|_{L^{p,q}}$$ would be equivalent to

which fails in general. However, since for all t > 0 we have

the estimate

(1.4.9)

where $$c_{p,q} = 2^{1/p}\max (1,2^{(1-q)/q})$$, is a consequence of (1.1.4). Also, if $$\|f\|_{L^{p,q}} = 0$$ then we must have f = 0 μ-a.e. Therefore, L p, q is a quasi-normed space for all p, q with 0 < p, q ≤ ∞. Is this space complete with respect to its quasi-norm? The next theorem answers this question.

Theorem 1.4.11.

Let (X,μ) be a measure space. Then for all $$0 < p,q \leq \infty $$ , the spaces L p,q (X,μ) are complete with respect to their quasi-norm and they are therefore quasi-Banach spaces.

Proof.

We consider only the case $$p < \infty $$. First we note that convergence in L p, q implies convergence in measure. When $$q = \infty $$, this is proved in Proposition 1.1.9. When $$q < \infty $$, in view of Proposition 1.4.5 (16) and (1.4.7), it follows that

for all $$f \in L^{p,q}$$, and from this it follows that convergence in L p, q implies convergence in measure.

Now let {f n } be a Cauchy sequence in L p, q . Then {f n } is Cauchy in measure, and hence it has a subsequence $$\{f_{n_{k}}\}$$ that converges almost everywhere to some f by Theorem 1.1.13. Fix k ₀ and apply property (9) in Proposition 1.4.5. Since $$\vert f - f_{n_{k_{ 0}}}\vert =\lim _{k\rightarrow \infty }\vert f_{n_{k}} - f_{n_{k_{ 0}}}\vert $$, it follows that

(1.4.10)

Raise (1.4.10) to the power q, multiply by t q∕p , integrate with respect to dt∕t over $$(0,\infty )$$, and apply Fatou’s lemma to obtain

(1.4.11)

Now let $$k_{0} \rightarrow \infty $$ in (1.4.11) and use the fact that {f n } is Cauchy to conclude that $$f_{n_{k}}$$ converges to f in L p, q . It is a general fact that if a Cauchy sequence has a convergent subsequence in a quasi-normed space, then the sequence is convergent to the same limit. It follows that f n converges to f in L p, q . □ 

Remark 1.4.12.

It can be shown that the spaces L p, q are normable when p, q are bigger than 1; see Exercise 1.4.3. Therefore, these spaces can be normed to become Banach spaces.

It is well known that finitely simple functions are dense in L p of any measure space, when $$0 < p < \infty $$. It is natural to ask whether finitely simple functions are also dense in L p, q . This is in fact the case when $$q\neq \infty $$.

Theorem 1.4.13.

Finitely simple functions are dense in L p,q (X,μ) when $$0 < q < \infty $$ .

Proof.

Let f ∈ L p, q (X, μ). Assume without loss of generality that f ≥ 0. Since f lies in $$L^{p,q} \subseteq L^{p,\infty }$$ we have $$\mu (\{f >\varepsilon \})^{1/p}\varepsilon \leq \| f\|_{L^{p,q}} < \infty $$ for every $$\varepsilon > 0$$ and consequently for any A > 0, μ({f > A}) is finite and tends to zero as $$A \rightarrow \infty $$. Thus for every $$n = 1,2,3,\ldots$$, there is an A n  > 0 such that $$\mu (\{f > A_{n}\}) < 2^{-n}$$.

For each $$n = 1,2,3,\ldots$$ define the function

Then $$\varphi _{n}$$ is supported in the set $$\{2^{-n} < f \leq A_{n}\}$$ which has finite μ measure, thus $$\varphi _{n}$$ is finitely simple and satisfies

for every $$x \in \{ x \in X:\,\, 2^{-n} < f(x) \leq A_{n}\}$$. It follows that

which implies that $$(f -\varphi _{n})^{{\ast}}(t) \leq 2^{-n}$$ for t ≥ 2−n . Thus

Since $$(f -\varphi _{n})^{{\ast}}(t) \leq f^{{\ast}}(t)$$, an application of the Lebesgue dominated convergence theorem gives that $$\|\varphi _{n} - f\|_{L^{p,q}} \rightarrow 0$$ as $$n \rightarrow \infty $$. □ 

Remark 1.4.14.

One may wonder whether simple functions are dense in $$L^{p,\infty }$$. This turns out to be false for all $$0 < p \leq \infty $$. However, countable linear combinations of characteristic functions of sets with finite measure are dense in $$L^{p,\infty }(X,\mu )$$. We call such functions countably simple. See Exercise 1.4.4 for details.

1.4.3 Duals of Lorentz Spaces

Given a quasi-Banach space Z with norm $$\|\,\cdot \,\|_{Z}$$, its dual Z ∗ is defined as the space of all continuous linear functionals T on Z equipped with the norm

Observe that the dual of a quasi-Banach space is always a Banach space.

We are now considering the following question: What are the dual spaces (L p, q )∗ of L p, q ? The answer to this question presents some technical difficulties for general measure spaces. In this exposition we restrict our attention to $$\sigma$$-finite nonatomic measure spaces, where the situation is simpler.

Definition 1.4.15.

A measurable subset A of a measure space (X, μ) is called an atom if μ(A) > 0 and every measurable subset B of A has measure either equal to zero or equal to μ(A). A measure space (X, μ) is called nonatomic if it contains no atoms. In other words, X is nonatomic if and only if for any $$A \subseteq X$$ with μ(A) > 0, there exists a proper subset $$B \subsetneq A$$ with μ(B) > 0 and μ(A ∖ B) > 0.

For instance, $$\mathbf{R}$$ with Lebesgue measure is nonatomic, but any measure space with counting measure is atomic. Nonatomic spaces have the property that every measurable subset of them with strictly positive measure contains subsets of any given measure smaller than the measure of the original subset. See Exercise 1.4.5.

Theorem 1.4.16.

Suppose that (X,μ) is a nonatomic σ-finite measure space. Then

Proof.

Since X is $$\sigma$$-finite, we have $$X =\bigcup _{ N=1}^{\infty }K_{N}$$, where K N is an increasing sequence of sets with $$\mu (K_{N}) < \infty $$. Let $$\mathcal{A}$$ be the $$\sigma$$-algebra on which μ is defined and define $$\mathcal{A}_{N} =\{ A \cap K_{N}:\,\, A \in \mathcal{A}\}$$. Given $$T \in (L^{p,q})^{{\ast}}$$, where $$0 < p,q < \infty $$, for each $$N = 1,2,\ldots$$, consider the measure $$\sigma _{N}(E) = T(\chi _{E})$$ defined on $$\mathcal{A}_{N}$$. Since $$\sigma _{N}$$ satisfies $$\vert \sigma _{N}(E)\vert \leq (p/q)^{1/q}\|T\|\mu (E)^{1/p}$$, it follows that $$\sigma _{N}$$ is absolutely continuous with respect to μ restricted on $$\mathcal{A}_{N}$$. By the Radon–Nikodym theorem (see [153] (19.36)), there exists a unique (up to a set of μ-measure zero) complex-valued measurable function g N which satisfies $$\int _{K_{N}}\vert g_{N}\vert \,d\mu < \infty $$ such that

(1.4.12)

for all f in $$L^{1}(K_{N},\mathcal{A}_{N},\sigma _{N})$$. Since $$\sigma _{N} =\sigma _{N+1}$$ on $$\mathcal{A}_{N}$$, it follows that $$g_{N} = g_{N+1}$$ μ-a.e. on K N and hence there is a well-defined measurable function g on X that coincides with each g N on K N . But the linear functionals f↦T(f) and $$f\mapsto \int _{K_{N}}f\,d\sigma _{N}$$ coincide on simple functions supported in K N and therefore they must be equal on $$L^{1}(K_{N},\mathcal{A}_{N},\sigma _{N}) \cap L^{p,q}(X,\mu )$$ by density; consequently, (1.4.12) is also equal to T(f) for f in $$L^{1}(K_{N},\mathcal{A}_{N},\sigma _{N}) \cap L^{p,q}(X,\mu )$$.

Note that if $$f \in L^{\infty }(K_{N},\mu )$$, then $$f \in L^{p,q}(K_{N},\mu )$$ and also in $$L^{\infty }(K_{N},\sigma _{N})$$, which is contained in $$L^{1}(K_{N},\mathcal{A}_{N},\sigma _{N})$$. It follows from (1.4.12) and the preceding discussion that

(1.4.13)

for every $$f \in L^{\infty }(K_{N})$$. We have now proved that for every linear functional T on $$L^{p,q}(X,\mu )$$ with $$0 < p,q < \infty $$ there is a function g satisfying $$\int _{K_{N}}\vert g\vert \,d\mu < \infty $$ for all $$N = 1,2,\ldots$$ such that (1.4.13) holds for all $$f \in L^{\infty }(K_{N})$$.

We now examine each case (i)–(viii) separately.

(i) We consider the case 0 < p < 1. Let $$f =\sum _{n}a_{n}\chi _{E_{n}}$$ be a finitely simple function on X (which is taken to be countably simple when $$q = \infty $$). Since X is nonatomic, we split each E n as a union of m disjoint sets E j, n , $$j = 1,2,\ldots,m$$, each having measure $$m^{-1}\mu (E_{n})$$. Let $$f_{j} =\sum _{n}a_{n}\chi _{E_{j,n}}$$. We see that $$\|f_{j}\|_{L^{p,q}} = m^{-1/p}\|f\|_{L^{p,q}}$$. Now if $$T \in (L^{p,q})^{{\ast}}$$, it follows that

Let $$m \rightarrow \infty $$ and use that p < 1 to obtain that T = 0.

(ii) We now consider the case p = 1 and 0 < q ≤ 1. Clearly, every $$h \in L^{\infty }$$ gives a bounded linear functional on L 1, q , since

Conversely, suppose that $$T \in (L^{1,q})^{{\ast}}$$ where q ≤ 1. The function g given in (1.4.13) satisfies

for all $$E \subseteq K_{N}$$, and hence $$\vert g\vert \leq q^{-1/q}\|T\|$$ μ-a.e. on every K N ; see [307, Theorem 1.40 on p. 31] for a proof of this fact. It follows that $$\|g\|_{L^{\infty }} \leq q^{-1/q}\|T\|$$ and hence $$(L^{1,q})^{{\ast}} = L^{\infty }$$.

(iii) Let us now take p = 1, $$1 < q < \infty $$, and suppose that $$T \in (L^{1,q})^{{\ast}}$$. Then

(1.4.14)

where g is the function in (1.4.13) and $$f \in L^{\infty }(K_{N})$$. We will show that g = 0 a.e. Suppose that $$\vert g\vert \geq \delta$$ on some set E ₀ with μ(E ₀) > 0. Then there exists N such that $$\mu (E_{0} \cap K_{N}) > 0$$. Let $$f = \overline{g}\vert g\vert ^{-2}\chi _{E_{0}\cap K_{N}}h\chi _{h\leq M}$$, where h is a nonnegative function. Then (1.4.14) implies for all $$h \geq 0$$ that

Letting $$M \rightarrow \infty $$, we obtain that $$L^{1,q}(E_{0} \cap K_{N})$$ is contained in $$L^{1}(E_{0} \cap K_{N})$$, but since the reverse inclusion is always valid, these spaces must be equal. Since X is nonatomic, this can’t happen; see Exercise 1.4.8 (d). Thus g = 0 μ-a.e. and T = 0.

(iv) In the case p = 1, $$q = \infty $$ an interesting phenomenon appears. Since every continuous linear functional on $$L^{1,\infty }$$ extends to a continuous linear functional on L 1, q for $$1 < q < \infty $$, it must necessarily vanish on all simple functions by part (iii). However, $$(L^{1,\infty })^{{\ast}}$$ contains nontrivial linear functionals; see [84], [85].

(v) We now take up the case $$1 < p < \infty $$ and 0 < q ≤ 1. Using Exercise 1.4.1 (b) and Proposition 1.4.10, we see that if f ∈ L p, q and $$h \in L^{p'\!,\infty }$$, then

thus every $$h \in L^{p',\infty }$$ gives rise to a bounded linear functional $$f\mapsto \int h\,f\,d\mu$$ on L p, q with norm at most $$C_{p,q}\|h\|_{L^{p'\!,\infty }}$$. Conversely, let $$T \in (L^{p,q})^{{\ast}}$$ where $$1 < p < \infty $$ and 0 < q ≤ 1. Let g satisfy (1.4.13) for all $$f \in L^{\infty }(K_{N})$$. Taking $$f = \overline{g}\vert g\vert ^{-1}\chi _{K_{N}\cap \{\vert g\vert >\alpha \}}$$ for $$\alpha > 0$$ and using that

we obtain that

Divide by $$\mu (K_{N} \cap \{\vert g\vert >\alpha \})^{\frac{1} {p} }$$, let $$N \rightarrow \infty $$, and take the supremum over $$\alpha > 0$$ to obtain that $$\|g\|_{L^{p'\!,\infty }} \leq (p/q)^{1/q}\|T\|$$.

(vi) Using Exercise 1.4.1 (b) and Hölder’s inequality, we obtain

thus every $$\varphi \in L^{p'\!,q'}$$ gives a bounded linear functional on L p, q with norm at most $$\|\varphi \|_{L^{p'\!,q'}}$$. Conversely, let T be in (L p, q )∗. By (1.4.13), T is given by integration against a locally integrable function g. It remains to prove that $$g \in L^{p'\!,q'}$$. We let $$g_{N,M} = g\,\chi _{K_{N}}\,\chi _{\vert g\vert \leq M}$$. Then $$(g_{N,M})^{{\ast}}\leq g^{{\ast}}$$ for all $$M,N = 1,2,\ldots$$ and $$(g_{N,M})^{{\ast}}\uparrow g^{{\ast}}$$ as $$M,N \rightarrow \infty $$ by Proposition 1.4.5 (4), (8).

For a bounded function f in L p, q (X) we have

(1.4.15)

where the first equality is a consequence of the fact that X is nonatomic (see Exercise 1.4.5 (d)). Using the result of Exercise 1.4.5 (b), pick a function f on X such that

(1.4.16)

noting that the preceding integral converges since $$(g_{N,M})^{{\ast}}(s) \leq M\,\chi _{[0,\mu (K_{N})]}(s)$$. It follows that $$f^{{\ast}}\leq c_{p,q}\,M^{q'-1}$$, which implies that f is bounded, and also that f ∗(t) = 0 when t > 2μ(K N ), which implies that f is supported in a set of measure at most 2μ(K N ); thus the function f defined in (1.4.16) is bounded and lies in L p, q (X).

We have the following calculation regarding the L p, q norm of f:

(1.4.17)

which is a consequence of Hardy’s second inequality in Exercise 1.2.8 with b = q∕p.

Using (1.4.15) and (1.4.17) we deduce that

(1.4.18)

On the other hand, we have

(1.4.19)

Combining (1.4.18) and (1.4.19), and using the fact that $$\|g_{N,M}\|_{L^{p'\!,q'}} < \infty $$, we obtain $$\|g_{N,M}\|_{L^{p'\!,q'}} \leq C(p,q)\|T\|$$. Letting $$N,M \rightarrow \infty $$ we deduce $$\|g\|_{L^{p'\!,q'}} \leq C(p,q)\|T\|$$ and this proves the reverse inequality required to complete case (vi).

(vii) For a complete characterization of this space, we refer to [83].

(viii) The dual of $$L^{\infty } = L^{\infty,\infty }$$ can be identified with the set of all bounded finitely additive set functions; see [99]. □ 

Remark 1.4.17.

Some parts of Theorem 1.4.16 are false if X is atomic. For instance, the dual of $$\ell^{p}(\mathbf{Z})$$ contains $$\ell^{\infty }$$ when 0 < p < 1 and thus it is not equal to {0}.

1.4.4 The Off-Diagonal Marcinkiewicz Interpolation Theorem

We now present the main result of this section, the off-diagonal extension of Marcinkiewicz’s interpolation theorem (Theorem 1.3.2). For a measure space (X, μ), let S(X) be the space of finitely simple functions on X and S ₀ ⁺(X) be the subset of S(X) of functions of the form

where m < n are integers and A i are subsets of X of finite measure. The sets A i are not required to be different nor disjoint; consequently, the sum of two elements in S ₀ ⁺(X) also belong to S ₀ ⁺(X). We define $$S_{0}^{real}(X) = S_{0}^{+}(X) - S_{0}^{+}(X)$$ be the space of all functions of the form $$f_{1} - f_{2}$$, where $$f_{1},f_{2}$$ lie in S ₀ ⁺(X) and S ₀(X) be the space of functions of the form $$h_{1} + ih_{2}$$, where $$h_{1},h_{2}$$ lie in $$S_{0}^{real}(X)$$.

An operator T defined on S ₀(X) is called quasi-linear if there is a $$K \geq 1$$ such that

for all $$\lambda \in \mathbf{C}$$ and all functions f, g in S ₀(X). If K = 1, then T is called sublinear .

Definition 1.4.18.

Let T be a linear operator defined on the space of finitely simple functions S(X) on a measure space (X, μ) and let $$0 < p,q \leq \infty $$. We say that T is of restricted weak type (p, q) if

(1.4.20)

for all measurable subsets A of X with finite measure. Estimates of the form (1.4.20) are called restricted weak type estimates.

It is important to observe that if an operator is of restricted weak type $$(p_{0},q_{0})$$ and of restricted weak type $$(p_{1},q_{1})$$, then it is of restricted weak type (p, q), where the indices are as in (1.4.23). It will be a considerable effort to extend the latter estimate to all functions in S ₀(X). The next theorem addresses this extension.

Theorem 1.4.19.

Let $$0 < r \leq \infty $$, $$0 < p_{0}\neq p_{1} \leq \infty $$ , and $$0 < q_{0}\neq q_{1} \leq \infty $$ and let (X,μ), (Y,ν) be $$\sigma$$ -finite measure spaces. Let T be a quasi-linear operator defined on the space of simple functions on X and taking values in the set of measurable functions on Y. Assume that for some $$M_{0},M_{1} < \infty $$ the following restricted weak type estimates hold:

(1.4.21)

(1.4.22)

for all measurable subsets A of X with $$\mu (A) < \infty $$ . Fix 0 < θ < 1 and let

(1.4.23)

Then there exists a constant $$C_{{\ast}}(p_{0},q_{0},p_{1},q_{1},K,r,\theta ) < \infty $$ such that for all functions f in S ₀ (X) we have

(1.4.24)

Additionally, if $$0 < p,r < \infty $$ and if T is linear (or sublinear with nonnegative values), then it admits a unique bounded extension from L p,r (X) to L q,r (Y,ν) such that(1.4.24)holds for all f in L p,r .

Before we give the proof of Theorem 1.4.19, we state and prove a lemma that is interesting on its own.

Lemma 1.4.20.

Let $$0 < p < \infty $$ and $$0 < q \leq \infty $$ and let (X,μ), (Y,ν) be $$\sigma$$ -finite measure spaces. Let T be a quasi-linear operator defined on S(X) and taking values in the set of measurable functions on Y. Suppose that there exists a constant M > 0 such that for all measurable subsets A of X of finite measure we have

(1.4.25)

Then for all α with $$0 <\alpha <\min (q, \frac{\log 2} {\log 2K})$$ there exists a constant $$C(p,q,K,\alpha ) > 0$$ such that for all functions f in S ₀ (X) we have the estimate

(1.4.26)

where

Proof.

A function f in S ₀(X) can be written as $$f = h_{1} - h_{2} + i(h_{3} - h_{4})$$, where h j are in $$S_{0}^{+}(X)$$. We write $$f = f_{1} - f_{2} + i(f_{3} - f_{4})$$, where $$f_{1} =\max (h_{1} - h_{2},0)$$, $$f_{2} =\max (-(h_{1} - h_{2}),0)$$, $$f_{3} =\max (h_{3} - h_{4},0)$$, and $$f_{4} =\max (-(h_{3} - h_{4}),0)$$. We note that f j lie in S ₀ ⁺(X); indeed, if $$h_{1} =\sum _{\ell}2^{-\ell}\chi _{A_{\ell}}$$ and $$h_{2} =\sum _{k}2^{-k}\chi _{B_{k}}$$, where both sums are finite, then

Since the second sum is equal to $$\sum _{s=\ell+1}^{k}2^{-s}\chi _{A_{\ell}\cap B_{k}}$$, we obtain that $$h_{1} \in S_{0}^{+}(X)$$. Likewise we can show that $$f_{2},f_{3},f_{4}$$ lie in S ₀ ⁺(X). Moreover, we have $$f_{j} \leq \vert f\vert $$ and Proposition 1.4.5(4), yields

for all j = 1, 2, 3, 4. Suppose now that (1.4.26) holds for functions in S ₀ ⁺(X) with constant $$C'(p,q,\alpha )$$ in place of $$C(p,q,K,\alpha )$$. By the quasi-linearity of T we have

which proves (1.4.26) for all f in S ₀(X) with constant

We now prove (1.4.26) for functions in S ₀ ⁺(X) with constant $$C'(p,q,\alpha )$$ in place of $$C(p,q,K,\alpha )$$. It follows from the Aoki–Rolewicz theorem (Exercise 1.4.6) that for all $$N \in \mathbf{Z}^{+}$$ and for all $$f_{1},\ldots,f_{N}$$ in S ₀ ⁺(X) we have the pointwise inequality

(1.4.27)

where $$0 <\alpha \leq \alpha _{1}$$ and $$\alpha _{1}$$ satisfies the equation $$(2K)^{\alpha _{1}} = 2.$$ The second inequality in (1.4.27) is a simple consequence of the fact that $$\alpha \leq \alpha _{1}$$. Fix $$\alpha _{0}$$ with

This ensures that the quasi-normed space $$L^{q/\alpha,\infty }$$ is normable when $$\alpha \leq \alpha _{0}$$. In fact, since Y is $$\sigma$$-finite, Exercise 1.1.12 gives that the space $$L^{s,\infty }$$ is normable as long as s > 1 and there is an equivalent norm $$\vert \vert \vert f\vert \vert \vert _{L^{s,\infty }}$$ such that

Next we claim that for any nonnegative function f in S ₀ ⁺(X) we have

(1.4.28)

To show this, we write $$f =\sum _{ j=m}^{n}2^{-j}\chi _{S_{j}},$$ where m < n are integers, S j are subsets of X of finite measure for all $$j \in \{ m,m + 1,\ldots,n\}$$, μ(S m ) ≠ 0 and μ(S n ) ≠ 0. Setting $$B_{j} = S_{j} \cap A$$ we have

and $$2^{-m} \leq \| f\chi _{A}\|_{L^{\infty }(X)}$$.

We use (1.4.27) once and (1.4.3) twice in the following argument. We have

using $$B_{j} \subseteq A$$. Using that $$2^{-m} \leq \| f\chi _{A}\|_{L^{\infty }}$$ establishes (1.4.28).

We now apply (1.4.28) to obtain (1.4.26). For any $$f \in S_{0}^{+}(X)$$ we define measurable sets

(1.4.29)

and we note that these sets are pairwise disjoint. We may write the finitely simple function f as $$\sum _{j=1}^{n}a_{j}\chi _{E_{j}}$$, where $$0 < a_{j} < \infty $$, $$E_{1} \subseteq E_{2} \subseteq \cdots \subseteq E_{n}$$ and $$0 <\mu (E_{j}) < \infty $$ for $$j \in \{ 1,2,\ldots,n\}$$. Clearly, we have

. Thus, when $$t \in (\mu (E_{n}),\infty )$$, f ∗(t) vanishes, and when $$t \in (0,\mu (E_{1}))$$, $$f^{{\ast}}(t) =\sum _{ j=1}^{n}a_{j}$$ is a positive constant. So there exists $$N \in \mathbf{Z}^{+}$$ such that $$f^{{\ast}}(2^{k}) = 0$$ when k > N, and that $$f^{{\ast}}(2^{k})$$ is a positive constant when k < −N. This also implies that $$A_{k} = \varnothing $$ if | k |  > N and thus we express

Proposition 1.4.5(2) implies $$\mu (A_{k}) \leq d_{f}(f^{{\ast}}(2^{k+1})) \leq 2^{k+1}.$$ Using (1.4.27) we obtain

where we made use of (1.4.28) and in the last inequality, we used

This completes the proof of the required inequality for nonnegative functions in S ₀ ⁺(X) with constant

As noted, the constant in general is $$C(p,q,K,\alpha ) = 2^{4+ \frac{2} {p} }\,K^{3}\,C'(p,q,\alpha ).$$ □ 

We now proceed with the proof of Theorem 1.4.19.

Proof.

We assume that $$p_{0} < p_{1}$$, since if p ₀ > p ₁ we may simply reverse the roles of p ₀ and p ₁. We first consider the case $$p_{1},r < \infty $$. Lemma 1.4.20 implies that

(1.4.30)

for all f in S ₀(X), where $$m = \frac{1} {2}\min \big(q_{0},q_{1}, \frac{\log 2} {\log 2K},2r\big)$$, $$M_{0}' = C(p_{0},q_{0},K,m)M_{0}$$, $$M_{1}' = C(p_{1},q_{1},K,m)M_{1}$$, and $$C(p,q,K,\alpha )$$ is as in (1.4.26).

Fix a function f in S ₀(X). Split f = f t + f t as follows:

where $$\delta$$ is to be determined later and $$\gamma$$ is the following nonzero real number:

Using Exercise 1.1.10 we write

Observe the following facts

We summarize these observations in a couple of inequalities:

It follows from these inequalities that f t lies in $$L^{p_{0},m}$$ and f t lies in $$L^{p_{1},m}$$ for all t > 0.

The quasi-linearity of the operator T and (1.4.9) imply

(1.4.31)

It follows from (1.4.30) that

(1.4.32)

(1.4.33)

for all t > 0. Now use (1.4.32), (1.4.33), and the facts that

to estimate (1.4.31) by

which is the same as

(1.4.34)

(1.4.35)

Next, we change variables $$u =\delta t^{\gamma }$$ in the L r quasi-norm in (1.4.34) to obtain

where the last inequality is a consequence of Hardy’s inequality:

(1.4.36)

with $$g(s) = f^{{\ast}}(s)^{m}s^{m/p_{0}} \geq 0$$, p = r∕m ≥ 1 and b = r∕p ₀ − r∕p > 0. See Exercise 1.2.8 for the proof of (1.4.36).

Likewise, change variables $$u =\delta t^{\gamma }$$ in the L r quasi-norm of (1.4.35) to obtain

where the last inequality above is Hardy’s inequality:

(1.4.37)

with $$g(s) = f^{{\ast}}(s)^{m}s^{m/p_{1}} \geq 0$$, p = r∕m ≥ 1 and b = r∕p − r∕p ₁ > 0. See Exercise 1.2.8 for the proof of (1.4.37).

Combining these elements we deduce that given f in S ₀(X), we have that the expression in (1.4.34) plus the expression in (1.4.35) is at most

We choose $$\delta > 0$$ such that the two terms in the curly brackets above are equal. We deduce that

where θ is as in (1.4.23), i.e.,

This proves (1.4.24) in the case $$p_{1},r < \infty $$ with constant $$C_{{\ast}}(p_{0},q_{0},p_{1},q_{1},K,r,\theta )$$ equal to

where we recall that $$m = \frac{1} {2}\min \big(q_{0},q_{1}, \frac{\log 2} {\log 2K},2r\big)$$ and $$C(p_{j},q_{j},K,m)$$ is as in Lemma 1.4.20.

We now turn to the remaining cases $$p = \infty $$ or $$r = \infty $$. The restriction $$r < \infty $$ can be removed since $$C_{{\ast}}(p_{0},q_{0},p_{1},q_{1},K,r,\theta )$$ has a finite limit as $$r \rightarrow \infty $$ and, moreover, $$\|f\|_{L^{p,r}} =\| t^{1/p}f^{{\ast}}(t)\|_{L^{r}(dt/t)} \rightarrow \| t^{1/p}f^{{\ast}}(t)\|_{L^{\infty }(dt/t)} =\| f\|_{L^{p,\infty }}$$ as $$r \rightarrow \infty $$ and likewise $$\|T(f)\|_{L^{p,r}} \rightarrow \| T(f)\|_{L^{p,\infty }}$$ as $$r \rightarrow \infty $$; see Exercise 1.1.3 (a). The restriction $$p_{1} < \infty $$ can be removed as follows. Suppose that $$p_{1} = \infty $$. Then, since θ ∈ (0, 1) it follows that $$p < \infty $$ and we pick p ₂ > p and $$p_{2} < \infty $$. It is easy to see T satisfies the restricted weak type $$(p_{2},q_{2})$$ estimate

where

(1.4.38)

Using the result obtained when $$p_{1} < \infty $$ with p ₂ in place of p ₁ we obtain that

(1.4.39)

for all functions f in S ₀(X), where

(1.4.40)

Combining (1.4.38) and (1.4.40) and using (1.4.23) we deduce that $$\theta =\rho \varphi$$ and hence (1.4.39) yields (1.4.24) in the case where $$p_{1} = \infty $$. In this case we have

where $$\varphi$$ is any number satisfying $$1 >\varphi > 1 -\tfrac{p_{0}} {p}$$.

Finally, we address the last assertion of the theorem which claims that when $$p,r < \infty $$ and K = 1, the linear (or sublinear with nonnegative values) operator T initially defined on finitely simple functions has a unique bounded extension from L p, r (X) to L q, r (Y ), which also satisfies (1.4.24) (with the same constant). To obtain this conclusion, we will need to know that the space S ₀(X) is dense in L p, r (X) whenever $$0 < p,r < \infty $$. This is proved in Proposition 1.4.21 below. Assuming this proposition, we define the extension of T on L p, r (X) as follows:

Given f in L p, r (X) a sequence of functions f j in S ₀(X) that converge to f in L p, r (X), notice that the linearity (or the sublinearity and the fact that T(f) ≥ 0 for all f in S ₀(X)) implies

Using the boundedness of T from L p, r (X) to L q, r (Y ) we obtain that the sequence {T(f j )} j is Cauchy in L q, r (Y ) and by the completeness of this space, it must converge to a limit which we call $$\overline{T}(f)$$. We observe that $$\overline{T}(f)$$ is independent of the choice of the sequence $$\{f_{j}\}_{j}$$ that converges to f in L p, r . Moreover, one can show that $$\overline{T}$$ is linear (or sublinear with nonnegative values), $$\overline{T}(f)$$ coincides with T(f) on S ₀(X) and $$\overline{T}$$ is bounded from L p, r (X) to L q, r (Y ). Thus $$\overline{T}$$ is the unique bounded extension of T on the entire space L p, r (X). For details, see Exercise 1.4.17. □ 

Proposition 1.4.21.

For all $$0 < p,r < \infty $$ the space S ₀ (X) is dense in L p,r (X).

Proof.

Let $$f \in L^{p,r}(X)$$ and assume first that f ≥ 0. Using (1.4.5) and the fact that d f is decreasing on $$[0,\infty )$$, we obtain for any $$n \in \mathbf{Z}^{+}$$,

which implies that $$d_{f}(2^{-n}) < \infty.$$ Likewise, again in view of (1.4.5), we have

which implies that $$\lim _{n\rightarrow \infty }d_{f}(2^{n}) = 0$$. Thus, for any $$n \in \mathbf{Z}^{+}$$, there exists $$k_{n} \in \mathbb{N}$$ such that

Let $$E_{n} = \left \{x \in X:\; 2^{-n} < f(x) \leq 2^{k_{n}}\right \}$$ and note that $$\mu \left (E_{n}\right ) \leq d_{f}\left (2^{-n}\right ) < \infty $$ for each $$n \in \mathbf{Z}^{+}$$. We write $$f\chi _{E_{n}}$$ in binary expansion, that is, $$f\chi _{E_{n}}(x) =\sum _{ j=-k_{n}}^{\infty }d_{j}(x)2^{-j}$$, where d j (x) = 0 or 1. Let $$B_{j} =\{ x \in E_{n}:\, d_{j}(x) = 1\}$$. Then, $$\mu (B_{j}) \leq \mu (E_{n})$$ and $$f\chi _{E_{n}}$$ can be expressed as $$f\chi _{E_{n}} =\sum _{ j=-k_{n}}^{\infty }2^{-j}\chi _{B_{j}}$$.

Set $$f_{n} =\sum _{ j=-k_{n}}^{n}2^{-j}\chi _{B_{j}}$$. It is obvious that $$f_{n} \in S_{0}^{+}(X)$$ and $$f_{n} \leq f\chi _{E_{n}} \leq f$$. Observe that when x ∈ E n , we have

and that when $$x\notin E_{n}$$, we have f n (x) = 0 and $$f(x) > 2^{k_{n}}$$ or f(x) ≤ 2−n . It follows from these facts that

Hence, for $$2^{-n} \leq t < \infty $$ one has

This implies that $$\lim _{n\rightarrow \infty }(f - f_{n})^{{\ast}}(t) = 0$$ for all t ∈ (0, ∞). By Proposition 1.4.5 (5), (6), we obtain for all $$t \in (0,\infty )$$

The Lebesgue dominated convergence theorem gives $$\|f_{n} - f\|_{L^{p,r}(X)} \rightarrow 0$$ as n → ∞ which yields the required conclusion for nonnegative functions f in L p, r (X).

For a complex-valued function f ∈ L p, r (X), we write $$f = f_{1} - f_{2} + i(f_{3} - f_{4})$$, where f j are nonnegative functions in L p, r (X). By the preceding conclusion, there exist sequences $$\{f_{n}^{j}\}_{n\in \mathbf{Z}^{+}}$$, j = 1, 2, 3, 4, in $$S_{0}^{+}(X)$$ such that $$f_{n}^{j} \rightarrow f_{j}$$ in L p, r (X) as $$n \rightarrow \infty.$$ Set $$f_{n} = f_{n}^{1} - f_{n}^{2} + i(f_{n}^{3} - f_{n}^{4})$$. Using the fact that $$\|\cdot \|_{L^{p,r}(X)}$$ is a quasi-norm we obtain

which tends to zero as $$n \rightarrow \infty $$. This completes the proof. □ 

Corollary 1.4.22.

Let T be as in the statement of Theorem 1.4.19 and let $$0 < p_{0}\neq p_{1} \leq \infty $$ and $$0 < q_{0}\neq q_{1} \leq \infty $$ . If T is restricted weak type $$(p_{0},q_{0})$$ and $$(p_{1},q_{1})$$ with constants M ₀ and M ₁ , respectively, and for some 0 < θ < 1 we have

and p ≤ q, then T satisfies the strong type estimate

(1.4.41)

for all f in S ₀ (X). Moreover, if T is linear (or sublinear with nonnegative values), then it has a unique bounded extension from L p (X,μ) to L q (Y,ν) that satisfies estimate (1.4.41) for all f ∈ L p (X) with the constant $$C(p_{0},q_{0},p_{1},q_{1},\theta )$$ replaced by $$C(p_{0},q_{0},p_{1},$$ $$q_{1},\theta )2^{2/p}\max (1,2^{1/p-1})^{2}$$ .

Proof.

Since θ ∈ (0, 1) we must have $$p,q < \infty $$. Take r = q in Theorem 1.4.19 and note that $$\|f\|_{L^{p,r}} \leq \| f\|_{L^{p}}$$ since p ≤ q = r; see Proposition 1.4.10. The last assertion follows using Exercise 1.4.17. □ 

We now give examples to indicate why the assumptions p ₀ ≠ p ₁ and q ₀ ≠ q ₁ cannot be dropped in Theorem 1.4.19.

Example 1.4.23.

Let $$X = Y = \mathbf{R}$$ and

Then $$\alpha \vert \{x:\,\, \vert T(\chi _{A})(x)\vert >\alpha \} \vert ^{1/2} = 2^{1/2}\vert A \cap [0,1]\vert $$ and thus T is of restricted weak types (1, 2) and (3, 2). But observe that T does not map $$L^{2} = L^{2,2}$$ to L q, 2. Thus Theorem 1.4.19 fails if the assumption q ₀ ≠ q ₁ is dropped. The dual operator

satisfies $$\alpha \vert \{x:\,\, \vert S(\chi _{A})(x)\vert >\alpha \} \vert ^{1/q} \leq c\vert A\vert ^{1/2}$$ when q = 1 or 3, and thus it furnishes an example of an operator of restricted weak types (2, 1) and (2, 3) that is not L ² bounded. Thus Theorem 1.4.19 fails if the assumption $$p_{0}\neq p_{1}$$ is dropped.

As an application of Theorem 1.4.19, we give the following strengthening of Theorem 1.2.13.

We end this chapter with a corollary of the proof of Theorem 1.4.19.

Corollary 1.4.24.

Let 1 ≤ r < ∞, $$1 \leq p_{0}\neq p_{1} < \infty $$ , and $$0 < q_{0}\neq q_{1} \leq \infty $$ and let (X,μ) and (Y,ν) be $$\sigma$$ -finite measure spaces. Let T be a quasi-linear operator defined on $$L^{p_{0}}(X) + L^{p_{1}}(X)$$ and taking values in the set of measurable functions on Y. Assume that for some $$M_{0}',M_{1}' < \infty $$ the following estimates hold for j = 0,1

(1.4.42)

for all functions $$f \in L^{p_{j}}(X)$$ . Fix 0 < θ < 1 and let

(1.4.43)

Then there exists a constant $$C_{{\ast}}(p_{0},q_{0},p_{1},q_{1},K,r,\theta ) < \infty $$ such that for all functions f in L p (X) we have

(1.4.44)

Proof.

Since L p (X) is contained in the sum $$L^{p_{0}}(X) + L^{p_{1}}(X)$$, the operator T is well defined on L p (X). Hypothesis (1.4.42) implies that (1.4.30) holds for all $$f \in L^{p_{j},1}$$. Repeat the proof of Theorem 1.4.19 starting from (1.4.30) fixing a function f in L p (X), m = 1 and r = p. We obtain the required conclusion.

Theorem 1.4.25.

(Young’s inequality for weak type spaces) Let G be a locally compact group with left Haar measure λ that satisfies (1.2.12) for all measurable subsets A of G. Let 1 < p,q,r < ∞ satisfy

(1.4.45)

Then there exists a constant B p,q,r > 0 such that for all f in L p (G) and g in L r,∞ (G) we have

(1.4.46)

Proof.

We fix 1 < p, q < ∞. Since p and q range in an open interval, we can find p ₀ < p < p ₁, $$q_{0} < q < q_{1}$$, and 0 < θ < 1 such that (1.4.23) and (1.4.45) hold. Let T(f) = f ∗ g, defined for all functions f on G. By Theorem 1.2.13, T extends to a bounded operator from $$L^{p_{0}}$$ to $$L^{q_{0},\infty }$$ and from $$L^{p_{1}}$$ to $$L^{q_{1},\infty }$$. It follows from the Corollary 1.4.24 that T extends to a bounded operator from L p (G) to L q (G). Notice that since G is locally compact, $$(G,\lambda )$$ is a $$\sigma$$-finite measure space and for this reason, we were able to apply Corollary 1.4.24. □ 

Exercises

1.4.1. (a) Let g be a nonnegative integrable function on a measure space (X, μ) and let A be a measurable subset of X. Prove that

(b) (G. H. Hardy and J. E. Littlewood) For f and g measurable on a $$\sigma$$-finite measure space (X, μ), prove that

Compare this result to the classical Hardy–Littlewood result asserting that for $$a_{j},b_{j} > 0$$, the sum $$\sum _{j}a_{j}b_{j}$$ is greatest when both a j and b j are rearranged in decreasing order (for this see [148, p. 261]).

1.4.2. Let (X, μ) be a measure space. Prove that if $$f \in L^{q_{0},\infty }(X) \cap L^{q_{1},\infty }(X)$$ for some $$0 < q_{0} < q_{1} \leq \infty $$, then $$f \in L^{q,s}(X)$$ for all $$0 < s \leq \infty $$ and $$q_{0} < q < q_{1}$$.

1.4.3. ([164]) Given $$0 < p,q < \infty $$, fix an r = r(p, q) > 0 such that r ≤ 1, r ≤ q and r < p. Let (X, μ) be a measure space. For t < μ(X) define

while for t ≥ μ(X) (if $$\mu (X) < \infty $$) let

Also define

(The function f ∗∗ and the functional $$f \rightarrow \vert \vert \vert f\vert \vert \vert _{L^{p,q}}$$ depend on r.)

(a) Prove that the inequality $$(((f + g)^{{\ast}{\ast}})(t))^{r} \leq (f^{{\ast}{\ast}}(t))^{r} + (g^{{\ast}{\ast}}(t))^{r}$$ is valid for all t ≥ 0. Since r ≤ q, conclude that the functional $$f \rightarrow \vert \vert \vert f\vert \vert \vert _{L^{p,q}}^{r}$$ is subadditive and hence it is a norm when r = 1 (this is possible only if p > 1).

(b) Show that for all f we have

(c) Conclude that L p, q (X) is metrizable and normable when 1 < p, q < ∞.

1.4.4. Show that on a measure space (X, μ) the set of countable linear combinations of simple functions is dense in $$L^{p,\infty }(X)$$.

(b) Prove that finitely simple functions are not dense in $$L^{p,\infty }(\mathbf{R})$$ for any $$0 < p \leq \infty $$.

[Hint: Part (b): Show that the function $$h(x) = x^{-1/p}\chi _{x>0}$$ cannot be approximated in $$L^{p,\infty }$$ by a sequence of finitely simple functions. Given a finitely simple function s which is nonzero on a set A with | A |  > 0, show that $$\|s - h\|_{L^{p,\infty }} \geq \sup _{0<\lambda <\vert A\vert ^{1/p}}\lambda (\lambda ^{-p}$$ $$-\vert A\vert )^{1/p} = 1$$.]

1.4.5. Let (X, μ) be a nonatomic measure space.

(a) If $$A_{0} \subseteq A_{1} \subseteq X$$, $$0 <\mu (A_{1}) < \infty $$, and $$\mu (A_{0}) \leq t \leq \mu (A_{1})$$, show that there exists an $$E_{t} \subseteq A_{1}$$ with μ(E t ) = t.

(b) Given a nonnegative continuous and decreasing function $$\varphi$$ on $$[0,\infty )$$ such that $$\varphi (t) = 0$$ whenever $$t \geq \mu (X)$$, prove that there exists a measurable function f on X with $$f^{{\ast}}(t) =\varphi (t)$$ for all t > 0.

(c) Given $$A \subseteq X$$ with $$0 <\mu (A) < \infty $$ and g an integrable function on X, show that there exists a subset $$\widetilde{A}$$ of X with $$\mu (\widetilde{A}) =\mu (A)$$ such that

(d) If X is $$\sigma$$-finite, $$f \in L^{\infty }(X)$$, and g ∈ L ¹(X), prove that

where the supremum is taken over all functions h on X equidistributed with f.

[Hint: Part (a): Reduce matters to the situation in which $$A_{0} = \varnothing $$. Consider first the case that for all A ⊆ X there exists a subset B of X satisfying $$\frac{1} {10}\,\mu (A) \leq \mu (B) \leq \frac{9} {10}\,\mu (A)$$. Then we can find subsets of A ₁ of measure in any arbitrarily small interval, and by continuity the required conclusion follows. Next consider the case in which there is a subset A ₁ of X such that every $$B \subseteq A_{1}$$ satisfies $$\mu (B) < \frac{1} {10}\,\mu (A_{1})$$ or $$\mu (B) > \frac{9} {10}\,\mu (A_{1})$$. Without loss of generality, normalize μ so that μ(A ₁) = 1. Let $$\mu _{1} =\sup \{\mu (C):\,\, C \subseteq A_{1},\,\,\mu (C) < \frac{1} {10}\}$$ and pick $$B_{1} \subseteq A_{1}$$ such that $$\frac{1} {2}\,\mu _{1} \leq \mu (B_{1}) \leq \mu _{1}$$. Set $$A_{2} = A_{1}\setminus B_{1}$$ and define $$\mu _{2} =\sup \{\mu (C):\,\, C \subseteq A_{2},\,\mu (C) < \frac{1} {10}\}$$. Continue in this way and define sets $$A_{1} \supseteq A_{2} \supseteq A_{3} \supseteq \cdots $$ and numbers $$\frac{1} {10} \geq \mu _{1} \geq \mu _{2} \geq \mu _{3} \geq \cdots $$. If $$C \subseteq A_{n+1}$$ with $$\mu(C )<\frac{1}{10}$$, then $$C \cup B_{n} \subseteq A_{n}$$ with $$\mu (C \cup B_{n}) < \frac{1} {5} < \frac{9} {10}$$, and hence by assumption we must have $$\mu (C \cup B_{n}) < \frac{1} {10}$$. Conclude that $$\mu _{n+1} \leq \frac{1} {2}\mu _{n}$$ and that $$\mu (A_{n}) \geq \frac{4} {5}$$ for all $$n = 1,2,\ldots$$. Then the set $$\bigcap _{n=1}^{\infty }A_{n}$$ must be an atom. Part (b): First show that when d is a simple right continuous decreasing function on $$[0,\infty )$$ there exists a measurable f on X such that f ∗ = d. For general continuous functions, use approximation. Part (c): Let t = μ(A) and define $$A_{1} =\{ x:\,\, \vert g(x)\vert > g^{{\ast}}(t)\}$$ and $$A_{2} =\{ x:\,\, \vert g(x)\vert \geq g^{{\ast}}(t)\}$$. Then $$A_{1} \subseteq A_{2}$$ and $$\mu (A_{1}) \leq t \leq \mu (A_{2})$$. Pick $$\widetilde{A}$$ such that $$A_{1} \subseteq \widetilde{ A} \subseteq A_{2}$$ and $$\mu (\widetilde{A}) = t =\mu (A)$$ by part (a). Then $$\int _{\widetilde{A}}g\,d\mu =\int _{X}g\chi _{\widetilde{A}}\,d\mu =\int _{ 0}^{\infty }(g\chi _{\widetilde{A}})^{{\ast}}\,ds =\int _{ 0}^{\mu (\widetilde{A})}g^{{\ast}}(s)\,ds.$$ Part (d): Reduce matters to functions $$f,g \geq 0$$. Let $$f =\sum _{ j=1}^{N}a_{j}\chi _{A_{j}}$$ where $$a_{1} > a_{2} >\ldots > a_{N} > 0$$ and the A j are pairwise disjoint. Write f as $$\sum _{j=1}^{N}b_{j}\chi _{B_{j}}$$, where $$b_{j} = (a_{j} - a_{j+1})$$ and $$B_{j} = A_{1} \cup \ldots \cup A_{j}$$. Pick $$\widetilde{B}_{j}$$ as in part (c). Then $$\widetilde{B}_{1} \subseteq \ldots \subseteq \widetilde{ B}_{N}$$ and the function $$f_{1} =\sum _{ j=1}^{N}b_{j}\chi _{\widetilde{B}_{j}}$$ has the same distribution function as f. It follows from part (c) that $$\int _{X}f_{1}g\,d\mu =\int _{ 0}^{\infty }f^{{\ast}}(s)g^{{\ast}}(s)\,ds.$$ The case of a general $$f \in L^{\infty }(X)$$ follows by approximation by finitely simple functions.]

1.4.6. ([7], [297]) Let $$K \geq 1$$ and let $$\|\,\cdot \,\|$$ be a nonnegative functional on a vector space X that satisfies

for all x, y ∈ X. For a fixed $$\alpha \leq 1$$ satisfying $$(2K)^{\alpha } = 2$$ show that

for all $$n = 1,2,\ldots$$ and all x ₁, x ₂, $$\ldots$$, x n in X. This inequality is referred to as the Aoki-Rolewicz theorem.

[Hint: Quasi-linearity implies that $$\|x_{1} +\ldots +x_{n}\| \leq \max _{1\leq j\leq n}[(2K)^{j}\|x_{j}\|]$$ for all $$x_{1},\ldots,x_{n}$$ in X (use that K ≥ 1). Define $$H:\,\, X \rightarrow \mathbf{R}$$ by setting H(0) = 0 and $$H(x) = 2^{j/\alpha }$$ if $$2^{j-1} <\| x\|^{\alpha } \leq 2^{j}$$. Then $$\|x\| \leq H(x) \leq 2^{1/\alpha }\|x\|$$ for all x ∈ X. Prove by induction that $$\|x_{1} +\ldots +x_{n}\|^{\alpha } \leq 2(H(x_{1})^{\alpha } +\ldots +H(x_{n})^{\alpha }).$$ Suppose that this statement is true when n = m. To show its validity for n = m + 1, without loss of generality assume that $$\|x_{1}\| \geq \| x_{2}\| \geq \ldots \geq \| x_{m+1}\|$$. Then $$H(x_{1}) \geq H(x_{2}) \geq \ldots \geq H(x_{m+1})$$. Assume that all the H(x j )’s are distinct. Then since $$H(x_{j})^{\alpha }$$ are distinct powers of 2, they must satisfy $$H(x_{j})^{\alpha } \leq 2^{-j+1}H(x_{1})^{\alpha }$$. Then

We now consider the case that $$H(x_{j}) = H(x_{j+1})$$ for some 1 ≤ j ≤ m. Then for some integer r we must have $$2^{r-1} <\| x_{j+1}\|^{\alpha } \leq \| x_{j}\|^{\alpha } \leq 2^{r}$$ and $$H(x_{j}) = 2^{r/\alpha }$$. Next note that

This implies

Now apply the inductive hypothesis to $$x_{1},\ldots,x_{j-1},x_{j} + x_{j+1},x_{j+1},\ldots,x_{m}$$ and use the previous inequality to obtain the required conclusion.]

1.4.7. (a) ([347]) Let (X, μ) and (Y, ν) be measure spaces. Let Z be a Banach space of complex-valued measurable functions on Y. Assume that Z is closed under absolute values and satisfies $$\|f\|_{Z} =\|\, \vert f\vert \,\|_{Z}$$. Suppose that T is a linear operator defined on the space of finitely simple functions on (X, μ) and taking values in Z. Suppose that for some constant A > 0 the following restricted weak type estimate

holds for some $$0 < p < \infty $$ and for all E measurable subsets of X of finite measure. Show that for all finitely simply functions f on X we have

Consequently T has a bounded extension from L p, 1(X) to Z.

(b) ([172]) As an application of part (a) prove that for any U, V measurable subsets of $$\mathbf{R}^{n}$$ with $$\vert U\vert,\vert V \vert < \infty $$ and any f measurable on U × V we have

[Hint: Part (a): Let $$f =\sum _{ j=1}^{N}a_{j}\chi _{E_{j}} \geq 0$$, where $$a_{1} > a_{2} >\ldots > a_{N} > 0$$, $$\mu (E_{j}) < \infty $$ pairwise disjoint. Let $$F_{j} = E_{1} \cup \ldots \cup E_{j}$$, B ₀ = 0, and $$B_{j} =\mu (F_{j})$$ for j ≥ 1. Write $$f =\sum _{ j=1}^{N}(a_{j} - a_{j+1})\chi _{F_{j}}$$, where a N+1 = 0. Then

where the penultimate equality follows by a summation by parts; see Appendix F.]

1.4.8. Let $$0 < p,q,\alpha,\beta < \infty $$. Also let $$0 < q_{1} < q_{2} < \infty $$.

(a) Show that the function $$f_{\alpha,\beta }(t) = t^{-\alpha }(\log t^{-1})^{-\beta }\chi _{[0,e^{-\beta /\alpha })}(t)$$ lies in $$L^{p,q}(\mathbf{R})$$ if and only if either $$p < 1/\alpha$$ or both $$p = 1/\alpha$$ and $$q > 1/\beta$$ hold. Conclude that the function $$t\mapsto t^{-1/p}(\log t^{-1})^{-1/q_{1}}\chi _{ [0,e^{-p/q_{1}})}(t)$$ lies in $$L^{p,q_{2}}(\mathbf{R})$$ but not in $$L^{p,q_{1}}(\mathbf{R})$$.

(b) Find a necessary and sufficient condition in terms of $$p,\alpha,\beta$$ for the function $$g_{\alpha,\beta }(t) = (1 + t)^{-\alpha }(\log (2 + t))^{-\beta }\chi _{[0,\infty )}$$ to lie in $$L^{p,q}(\mathbf{R})$$.

(c) Let ψ(t) be smooth decreasing function on $$[0,\infty )$$ and let F(x) = ψ( | x | ) for $$x$$ in $$\mathbf{R}^{n}$$, where | x | is the modulus of x. Show that $$F^{{\ast}}(t) = f((t/v_{n})^{1/n})$$, where v n is the volume of the unit ball. Use this formula to construct examples showing that $$L^{p,q_{1}}(\mathbf{R}^{n}) \subsetneq L^{p,q_{2}}(\mathbf{R}^{n})$$.

(d) On a general nonatomic measure space (X, μ) prove that there does not exist a constant $$C(p,q_{1},q_{2}) > 0$$ such that for all f in $$L^{p,q_{2}}(X)$$ the following is valid:

[Hint: Parts (a), (b): Use that $$f_{\alpha,\beta }$$ and $$g_{\alpha,\beta }$$ are equal to their decreasing rearrangements. Part (d): Use Exercise 1.4.5 (b) with $$\varphi (t) = g_{1/p,1/q_{1}}(t)$$.]

1.4.9. ([346]) Let $$L^{p}(\omega )$$ denote the space of all measurable functions f on $$\mathbf{R}^{n}$$ such that $$\|f\|_{L^{p}(\omega )}^{p} =\int _{\mathbf{R}^{n}}\vert f(x)\vert ^{p}\,\omega (x)\,dx < \infty $$, where $$0 <\omega < \infty $$ a.e. Let T be a sublinear operator that maps $$L^{p_{0}}(\omega _{0})$$ to $$L^{q_{0},\infty }(\omega )$$ and $$L^{p_{1}}(\omega _{1})$$ to $$L^{q_{1},\infty }(\omega )$$, where $$\omega _{0},\omega _{1},\omega$$ are positive functions and $$1 \leq p_{0} < p_{1} < \infty $$, $$0 < q_{0},q_{1} < \infty $$. Suppose that

Let $$\varOmega _{\theta } =\omega _{ 0}^{\frac{1-\theta } {p_{0}} p_{\theta }}\omega _{1}^{ \frac{\theta }{ p_{1}} p_{\theta }}$$. Show that T maps $$L^{p_{\theta }}\big(\varOmega _{\theta }\big) \rightarrow L^{q_{\theta },p_{\theta }}(\omega )\,.$$

[Hint: Define $$L(f) = (\omega _{1}/\omega _{0})^{ \frac{1} {p_{1}-p_{0}} }f$$ and observe that for each θ ∈ [0, 1], L maps $$L^{p_{\theta }}\big(\varOmega _{\theta }\big) \rightarrow L^{p_{\theta }}\big((\omega _{0}^{p_{1}}\omega _{1}^{-p_{0}})^{ \frac{1} {p_{1}-p_{0}} }\big)$$ isometrically. Then apply Corollary 1.4.24 to the sublinear operator T ∘ L ⁻¹.]

1.4.10. ([185], [349]) Let $$\lambda _{n}$$ be a sequence of positive numbers with $$\sum _{n}\lambda _{n} \leq 1$$ and $$\sum _{n}\lambda _{n}\log (\frac{1} {\lambda _{n}}) = K < \infty $$. Suppose all sequences are indexed by a fixed countable set.

(a) Let f n be a sequence of complex-valued functions in $$L^{1,\infty }(X)$$ with $$\|f_{n}\|_{L^{1,\infty }} \leq 1$$ uniformly in n. Prove that $$\sum _{n}\lambda _{n}f_{n}$$ lies in $$L^{1,\infty }(X)$$ with norm at most 2(K + 2). (This property is referred to as the logconvexity of $$L^{1,\infty }$$.)

(b) Let T n be a sequence of sublinear operators that map L ¹(X) to $$L^{1,\infty }(Y )$$ with norms $$\|T_{n}\|_{L^{1}\rightarrow L^{1,\infty }} \leq B$$ uniformly in n. Use part (a) to prove that $$\sum _{n}\lambda _{n}T_{n}$$ maps L ¹(X) to $$L^{1,\infty }(Y )$$ with norm at most 2B(K + 2).

(c) Given $$\delta > 0$$ pick $$0 <\varepsilon <\delta$$ and use the simple estimate

to obtain a simple proof of the statement in part (a) when $$\lambda _{n} = 2^{-\delta n}$$, $$n = 1,2,\ldots$$.

[Hint: Part (a): For fixed $$\alpha > 0$$, write $$f_{n} = u_{n} + v_{n} + w_{n}$$, where $$u_{n} = f_{n}\chi _{\vert f_{n}\vert \leq \frac{\alpha }{ 2} }$$, $$v_{n} = f_{n}\chi _{\vert f_{n}\vert > \frac{\alpha }{ 2\lambda _{n}} }$$, and $$w_{n} = f_{n}\chi _{ \frac{\alpha }{ 2} <\vert f_{n}\vert \leq \frac{\alpha }{ 2\lambda _{n}} }$$. Let $$u =\sum _{n}\lambda _{n}u_{n}$$, $$v =\sum _{n}\lambda _{n}v_{n}$$, and $$w =\sum _{n}\lambda _{n}w_{n}$$. Clearly $$\vert u\vert \leq \frac{\alpha } {2}$$. Also $$\{v\neq 0\} \subseteq \bigcup _{n}\{\vert f_{n}\vert > \frac{\alpha } {2\lambda _{ n}}\}$$; hence $$\mu (\{v\neq 0\}) \leq \frac{2} {\alpha }$$. Finally,

Using $$\mu (\{\vert u + v + w\vert >\alpha \}) \leq \mu (\{\vert u\vert >\alpha /2\}) +\mu (\{\vert v\vert \neq 0\}) +\mu (\{\vert w\vert >\alpha /2\}),$$ deduce the conclusion.]

1.4.11. Let $$\{f_{n}\}_{n}$$ be a sequence of measurable functions on a measure space (X, μ). Let $$0 < q,s \leq \infty $$.

(a) Suppose that $$f_{n} \geq 0$$ for all n. Show that

(b) Let g n  → g in L q, s as $$n \rightarrow \infty $$. Show that $$\|g_{n}\|_{L^{q,s}} \rightarrow \| g\|_{L^{q,s}}$$ as $$n \rightarrow \infty $$.

1.4.12. (a) Suppose that X is a quasi-Banach space and let X ∗ be its dual (which is always a Banach space). Prove that for all T ∈ X ∗ we have

(b) Now suppose that X is a Banach space. Use the Hahn–Banach theorem to prove that for every x ∈ X we have

Observe that this result may fail for quasi-Banach spaces. For example, if $$X = L^{1,\infty }$$, every linear functional on X ∗ vanishes on the set of simple functions.

(c) Let $$1 < p < \infty $$, X = L p, 1(Y ), and $$X^{{\ast}} = L^{p',\infty }(Y )$$, where (Y, μ) is nonatomic σ-finite measure space. Conclude that

1.4.13. Let $$0 < p,q < \infty $$. Prove that any function in $$L^{p,q}(X,\mu )$$ can be written as

where f n is a function bounded by $$2^{-n/p}$$, supported on a set of measure 2 n , and the sequence $$\{c_{k}\}_{k}$$ lies in ℓ q and satisfies

[Hint: Let $$c_{n} = 2^{n/p}f^{{\ast}}(2^{n})$$, $$A_{n} =\{ x: f^{{\ast}}(2^{n+1}) < \vert f(x)\vert \leq f^{{\ast}}(2^{n}))\}$$, and $$f_{n} = c_{n}^{-1}f\chi _{A_{n}}$$.]

1.4.14. (T. Tao) Let $$0 < p < \infty $$, $$0 <\gamma < 1$$, A, B > 0, and let f be a measurable function on a measure space (X, μ).

(a) Suppose that $$\|f\|_{L^{p,\infty }} \leq A$$. Then for every measurable set E of finite measure there exists a measurable subset E′ of E with $$\mu (E') \geq \gamma \,\mu (E)$$ such that f is integrable on E′ and

(b) Suppose that (X, μ) is a $$\sigma$$-finite measure space and that f has the property that for any measurable subset E of X with $$\mu (E) < \infty $$ there is a measurable subset E′ of E with $$\mu (E') \geq \gamma \,\mu (E)$$ such that f is integrable on E′ and

Then we have that $$\|f\|_{L^{p,\infty }} \leq B\,4^{1/p}\gamma ^{-1}\sqrt{2}$$.

(c) Conclude that if (X, μ) is a $$\sigma$$-finite measure space then

[Hint: Part (a): Take $$E' = E\setminus \{\vert f\vert > A(1-\gamma )^{-\frac{1} {p} }\mu (E)^{-\frac{1} {p} }\}$$. Part (b): Write $$X =\bigcup _{ n=1}^{\infty }X_{n}$$ with $$\mu (X_{n}) < \infty $$. Given $$\alpha > 0$$, note that the set $$\big\{\vert f\vert >\alpha \big\}$$ is contained in

Let $$E_{n}$$ be any of the preceding four sets intersected with X n , let E n ′ be a subset of it with measure at least $$\gamma \,\mu (E_{n})$$ as in the hypothesis. Then $$\big\vert \int _{E_{n}'}f\,d\mu \big\vert \geq \frac{\alpha } {\sqrt{2}}\gamma \,\mu (E_{n})$$, from which it follows that $$\alpha \mu (E_{n})^{1/p} \leq B\,\sqrt{2}\,\gamma ^{-1}$$, and let $$n \rightarrow \infty $$.]

1.4.15. Let T be a linear operator defined on the set of finitely simple functions on a σ-finite measure space (X, μ) and taking values in the set of measurable functions on a $$\sigma$$-finite measure space (Y, ν) and T t be a linear operator defined on the set of finitely simple functions on (Y, ν) and taking values in the set of measurable functions of (X, μ). Suppose that for all A subsets of X and B subsets of Y of finite measure we have

and that T and T t are related via the “transpose identity”

Assume that whenever $$\mu (A_{n}) +\nu (B_{n}) \rightarrow 0$$ as $$n \rightarrow \infty $$, we have $$\varLambda (A_{n},B_{n}) \rightarrow 0$$. Suppose that T and T t are restricted weak type (1,1) operators, with constants C ₁ and C ₂, respectively. Show that, for all $$1 < p < \infty $$, T is of restricted weak type (p,p). Precisely, show that there exists a constant K p such that

for all measurable subsets A of X with $$\mu (A) < \infty $$.

[Hint: Suppose that $$C_{1}\mu (F) > C_{2}\nu (E)$$ and pick m so that $$C_{1}\mu (F) \sim 2^{m}C_{2}\nu (E).$$ Since T t is restricted weak type (1, 1) there is an $$F' \subseteq F$$ such that $$\mu (F') \geq \frac{1} {2}\mu (F)$$ and $$\vert \varLambda (F',E)\vert \leq 2C_{2}\nu (E).$$ Find by induction sets $$F^{(j)} \subseteq F\setminus (F' \cup \ldots \cup F^{(j-1)})$$ such that $$\mu (F^{(j)}) \geq \frac{1} {2}\mu (F\setminus (F' \cup \ldots \cup F^{(j-1)}))$$ and $$\vert \varLambda (F^{(j)},E)\vert \leq 2C_{2}\nu (E),\,j = 1,2,\ldots,m\,.$$ Stop when $$F^{(m)} = F\setminus (F' \cup \ldots \cup F^{(m-1)})$$ satisfies $$C_{1}\mu (F^{(m)}) \leq C_{2}\nu (E).$$ Since T is restricted weak type (1, 1) there is a subset E′ of E such that $$\nu (E') \geq \frac{1} {2}\nu (E)$$ and $$\vert \varLambda (F^{(m)},E')\vert \leq 2C_{1}\mu (F^{(m)}) \leq 2C_{2}\nu (E).$$ Now write

from which it follows that

where F ₁ = F (m) and $$E_{1} = E\setminus E'$$. Note that the first term in the sum above is at most $$K_{p}'(C_{1}\mu (F))^{1/p}(C_{2}\nu (E))^{1/p'}$$ and that the identical estimate holds if the roles of E and F are reversed. Also observe that $$\mu (F_{1}) \leq \frac{1} {2}\mu (F)$$ and $$\nu (E_{1}) \leq \frac{1} {2}\nu (E)$$. Continuing this process we find sets $$(F_{n},E_{n})$$ with $$\mu (F_{n+1}) \leq \frac{1} {2}\mu (F_{n})$$ and $$\nu (E_{n+1}) \leq \frac{1} {2}\nu (E_{n})$$.Using $$\varLambda (F_{n},E_{n}) \rightarrow 0$$ as $$n \rightarrow \infty $$ we deduce that $$\vert \varLambda (F,E)\vert \leq 2K_{p}'(C_{1}\mu (F))^{1/p}(C_{2}\nu (E))^{1/p'}$$. Considering the sets $$E_{+} = E \cap \{ T(\chi _{F}) > 0\}$$ and $$E_{-} = E \cap \{ T(\chi _{F}) < 0\}$$, obtain that $$\int _{E}\big\vert T(\chi _{F})\big\vert \,d\nu \leq 4K_{p}'\big(C_{1}\mu (F)\big)^{\frac{1} {p} }\big(C_{2}\nu (E)\big)^{\frac{1} {p'} }$$ for all F and E measurable sets of finite measure. Exercise 1.1.12 (a) with r = 1 yields that $$\|T(\chi _{F})\|_{L^{p,\infty }} \leq 4K_{p}\,C_{1}^{1/p}C_{2}^{1/p'}\mu (F)^{1/p}$$. ]

1.4.16. ([35]) Let $$0 < p_{0} < p_{1} < \infty $$ and $$0 <\alpha,\beta,A,B < \infty $$. Suppose that a family of sublinear operators T k is of restricted weak type $$(p_{0},p_{0})$$ with constant $$A\,2^{-k\alpha }$$ and of restricted weak type (p ₁, p ₁) with constant $$B\,2^{k\beta }$$ for all $$k \in \mathbf{Z}$$. Show that there is a constant $$C = C(\alpha,\beta,p_{0},p_{1})$$ such that $$\sum _{k\in \mathbf{Z}}T_{k}$$ is of restricted weak type (p, p) with constant $$C\,A^{1-\theta }B^{\theta }$$, where $$\theta =\alpha /(\alpha +\beta )$$ and

[Hint: Estimate $$\mu (\{\vert T(\chi _{E})\vert >\lambda \})$$ by the sum $$\sum _{k\geq k_{0}}\mu (\{\vert T_{k}(\chi _{E})\vert > c\lambda 2^{\alpha '(k_{0}-k)}\})$$+ $$\sum _{k\leq k_{0}}\mu (\{\vert T_{k}(\chi _{E})\vert > c\lambda 2^{\beta '(k-k_{0})}\})$$, where c is a suitable constant and $$0 <\alpha ' <\alpha$$, $$0 <\beta ' <\beta$$. Apply the restricted weak type $$(p_{0},p_{0})$$ hypothesis on each term of the first sum, the restricted weak type $$(p_{1},p_{1})$$ hypothesis on each term of the second sum, and choose k ₀ to optimize the resulting expression.]

1.4.17. Let (X, μ), (Y, ν) be measure spaces, $$0 < p,r,q,s \leq \infty $$ and $$0 < B < \infty $$. Suppose that a sublinear operator T is defined on a dense subspace $$\mathcal{D}$$ of L p, r (X), takes values in the space of measurable functions of another measure space Y, and satisfies T(f) ≥ 0 for all f in $$\mathcal{D}$$. Assume that

for all $$\varphi$$ in $$\mathcal{D}$$. Prove that T admits a unique sublinear extension $$\overline{T}$$ on L p, r (X) such that

for all f ∈ L p, r (X).

[Hint: Given $$f \in L^{p,r}(X)$$ find a sequence of functions $$\varphi _{j}$$ in $$\mathcal{D}$$ such that $$\varphi _{j} \rightarrow f$$ in L p, r . Use the inequality $$\vert T(\varphi _{j}) - T(\varphi _{k})\vert \leq \vert T(\varphi _{j} -\varphi _{k})\vert \,,$$ to obtain that the sequence $$\{T(\varphi _{j})\}_{j}$$ is Cauchy in L q, s and thus it has a unique limit $$\overline{T}(f)$$ which is independent of the choice of sequence $$\varphi _{j}$$. Boundedness of $$\overline{T}$$ follows by density. To prove that $$\overline{T}$$ is sublinear use that convergence in L q, s implies convergence in measure and thus a subsequence of $$T(\varphi _{j})$$ converges ν-a.e. to $$\overline{T}(f)$$. Also use Exercise 1.4.11.]

HISTORICAL NOTES

The modern theory of measure and integration was founded with the publication of Lebesgue’s dissertation [214]; see also [215]. The theory of the Lebesgue integral reshaped the course of integration. The spaces L p ([a, b]), $$1 < p < \infty $$, were first investigated by Riesz [290], who obtained many important properties of them. A rigorous treatise of harmonic analysis on general groups can be found in the book of Hewitt and Ross [152]. The best possible constant $$C_{pqr}$$ in Young’s inequality $$\|f {\ast} g\|_{L^{r}(\mathbf{R}^{n})} \leq C_{pqr}\|f\|_{L^{p}(\mathbf{R}^{n})}\|g\|_{L^{q}(\mathbf{R}^{n})}$$, $$\frac{1} {p} + \frac{1} {q} = \frac{1} {r} + 1$$, $$1\! <\! p,q,r\! <\! \infty $$, was shown by Beckner [21] to be $$C_{pqr} = (B_{p}B_{q}B_{r'})^{n}$$, where $$B_{p}^{2} = p^{1/p}(p')^{-1/p'}$$.

Theorem 1.3.2 first appeared without proof in Marcinkiewicz’s brief note [240]. After his death in World War II, this theorem seemed to have escaped attention until Zygmund reintroduced it in [387]. This reference presents the more difficult off-diagonal version of the theorem, derived by Zygmund. Stein and Weiss [347] strengthened Zygmund’s theorem by assuming that the initial estimates are of restricted weak type whenever $$1 \leq p_{0},p_{1},q_{0},q_{1} \leq \infty $$. The extension of this result to the case $$0 < p_{0},p_{1},q_{0},q_{1} < 1$$ in Theorem 1.4.19 is due to the author. The critical Lemma 1.4.20 was suggested by Kalton. Improvements of these results, in particular, the appearance of the space S ₀(X) and the presence of the factor $$M_{0}^{1-\theta }M_{1}^{\theta }$$ in (1.4.24) appeared in Liang, Liu, and Yang [224]. Equivalence of restricted weak type (1, 1) and weak type (1, 1) properties for certain maximal multipliers was obtained by Moon [257]. The following partial converse of Theorem 1.2.13 is due to Stepanov [351]: If a convolution operator maps $$L^{1}(\mathbf{R}^{n})$$ to $$L^{q,\infty }(\mathbf{R}^{n})$$ for some $$1 < q < \infty $$ then its kernel must be in $$L^{q,\infty }$$.

The extrapolation result of Exercise 1.3.7 is due to Yano [380]; see also Zygmund [389, pp. 119–120] and the related work of Carro [56], Soria [330], and Tao [356].

The original version of Theorem 1.3.4 was proved by Riesz [293] in the context of bilinear forms. This version is called the Riesz convexity theorem, since it says that the logarithm of the function $$M(\alpha,\beta ) =\inf _{x,y}\big\vert \sum _{j=1}^{n}\sum _{k=1}^{m}a_{jk}x_{j}y_{k}\big\vert \,\|x\|_{\ell^{1/\alpha }}^{-1}\|y\|_{\ell^{1/\beta }}^{-1}$$ (where the infimum is taken over all sequences $$\{x_{j}\}_{j=1}^{n}$$ in $$\ell^{1/\alpha }\!$$ and $$\{y_{k}\}_{k=1}^{m}\!$$ in $$\ell^{1/\beta }$$) is a convex function of $$(\alpha,\beta )$$ in the triangle $$0 \leq \alpha,\beta \leq 1$$, $$\alpha +\beta \geq 1$$. Riesz’s student Thorin [360] extended this triangle to the unit square $$0 \leq \alpha,\beta \leq 1$$ and generalized this theorem by replacing the maximum of a bilinear form with the maximum of the modulus of an entire function in many variables. After the end of World War II, Thorin published his thesis [361], building the subject and giving a variety of applications. The original proof of Thorin was rather long, but a few years later, Tamarkin and Zygmund [354] gave a very elegant short proof using the maximum modulus principle in a more efficient way. Today, this theorem is referred to as the Riesz–Thorin interpolation theorem.

Calderón [42] elaborated the complex-variables proof of the Riesz–Thorin theorem into a general method of interpolation between Banach spaces. The complex interpolation method can also be defined for pairs of quasi-Banach spaces, although certain complications arise in this setting; however, the Riesz–Thorin theorem is true for pairs of L p spaces (with the “correct” geometric mean constant) for all $$0 < p \leq \infty $$ and also for Lorentz spaces. In this setting, duality cannot be used, but a well-developed theory of analytic functions with values in quasi-Banach spaces is crucial. We refer to the articles of Kalton [186] and [187] for details. Complex interpolation for sublinear maps is also possible; see the article of Calderón and Zygmund [47]. Interpolation for analytic families of operators (Theorem 1.3.7) is due to Stein [331]. The critical Lemma 1.3.8 used in the proof was previously obtained by Hirschman [154].

The fact that nonatomic measure spaces contain subsets of all possible measures is classical. An extension of this result to countably additive vector measures with values in finite-dimensional Banach spaces was obtained by Lyapunov [236]; for a proof of this fact, see Diestel and Uhl [95, p. 264]. The Aoki–Rolewicz theorem (Exercise 1.4.6) was proved independently by Aoki [7] and Rolewicz [297]. For a proof of this fact and a variety of its uses in the context of quasi-Banach spaces we refer to the book of Kalton, Peck, and Roberts [188].

Decreasing rearrangements of functions were introduced by Hardy and Littlewood [146]; the authors attribute their motivation to understanding cricket averages. The L p, q spaces were introduced by Lorentz in [232] and in [233]. A general treatment of Lorentz spaces is given in the article of Hunt [164]. The normability of the spaces L p, q (which holds exactly when $$1 < p \leq \infty $$ and $$1 \leq q \leq \infty $$) can be traced back to general principles obtained by Kolmogorov [199]. The introduction of the function f ∗∗, which was used in Exercise 1.4.3, to explicitly define a norm on the normable spaces L p, q is due to Calderón [42]. These spaces appear as intermediate spaces in the general interpolation theory of Calderón [42] and in that of Lions and Peetre [225]. The latter was pointed out by Peetre [275]. For a systematic study of the duals of Lorentz spaces we refer to Cwikel [83] and Cwikel and Fefferman [84], [85]. An extension of the Marcinkiewicz interpolation theorem to Lorentz spaces was obtained by Hunt [163]. Carro, Raposo, and Soria [57] provide a comprehensive presentation of the theory of Lorentz spaces in the context of weighted inequalities. For further topics on interpolation one may consult the books of Bennett and Sharpley [24], Bergh and Löfström [25], Sadosky [309], Kislyakov and Kruglyak [194], and Chapter 5 in Stein and Weiss [348].

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© Springer Science+Business Media New York 2014

Loukas GrafakosClassical Fourier AnalysisGraduate Texts in Mathematics24910.1007/978-1-4939-1194-3_2

2. Maximal Functions, Fourier Transform, and Distributions

Loukas Grafakos1 

(1)

Department of Mathematics, University of Missouri, Columbia, MO, USA

We have already seen that the convolution of a function with a fixed density is a smoothing operation that produces a certain average of the function. Averaging is an important operation in analysis and naturally arises in many situations. The study of averages of functions is better understood by the introduction of the maximal function which is defined as the largest average of a function over all balls containing a fixed point. Maximal functions are used to obtain almost everywhere convergence for certain integral averages and play an important role in this area, which is called differentiation theory. Although maximal functions do not preserve qualitative information about the given functions, they maintain crucial quantitative information, a fact of great importance in the subject of Fourier analysis.

Another important operation we study in this chapter is the Fourier transform, the father of all oscillatory integrals. This is as fundamental to Fourier analysis as marrow is to the human bone. It is a powerful transformation that carries a function from its spatial domain to its frequency domain. By doing this, it inverts the function’s localization properties. If applied one more time, then magically reproduces the function composed with a reflection. It changes convolution to multiplication, translation to modulation, and expanding dilation to shrinking dilation. Its decay at infinity encodes information about the local smoothness of the function. The study of the Fourier transform also motivates the launch of a thorough study of general oscillatory integrals. We take a quick look at this topic with emphasis on one-dimensional results.

Distributions suppy a mathematical framework for many operations that do not exactly qualify to be called functions. These operations found their mathematical place in the world of functionals applied to smooth functions (called test functions). These functionals also introduced the correct interpretation for many physical objects, such as the Dirac delta function. Distributions have become an indispensable tool in analysis and have enhanced our perspective.

2.1 Maximal Functions

Given a Lebesgue measurable subset A of $$\mathbf{R}^{n}$$, we denote by | A | its Lebesgue measure. For $$x \in \mathbf{R}^{n}$$ and r > 0, we denote by B(x, r) the open ball of radius r centered at x. We also use the notation $$aB(x,\delta ) = B(x,a\delta )$$, for a > 0, for the ball with the same center and radius $$a\delta$$. Given $$\delta > 0$$ and f a locally integrable function on $$\mathbf{R}^{n}$$, let

denote the average of | f | over the ball of radius $$\delta$$ centered at x.

2.1.1 The Hardy–Littlewood Maximal Operator

Definition 2.1.1.

Let f be a locally integrable function on $$\mathbf{R}^{n}$$. The function

is called the centered Hardy–Littlewood maximal function of f.

Obviously we have $$\mathcal{M}(f) = \mathcal{M}(\vert f\vert ) \geq 0$$; thus the maximal function is a positive operator. Information concerning cancellation of the function f is lost by passing to $$\mathcal{M}(f)$$. We show later that $$\mathcal{M}(f)$$ pointwise controls f (i.e., $$\mathcal{M}(f) \geq \vert f\vert $$ almost everywhere). Note that $$\mathcal{M}$$ maps $$L^{\infty }$$ to itself, that is, we have

Let us compute the Hardy–Littlewood maximal function of a specific function.

Example 2.1.2.

On $$\mathbf{R}$$, let f be the characteristic function of the interval [a, b]. For x ∈ (a, b), clearly $$\mathcal{M}(f) = 1$$. For x ≥ b, a simple calculation shows that the largest average of f over all intervals $$(x-\delta,x+\delta )$$ is obtained when $$\delta = x - a$$. Similarly, when x ≤ a, the largest average is obtained when $$\delta = b - x$$. Therefore,

Observe that $$\mathcal{M}(f)$$ has a jump at x = a and x = b equal to one-half that of f.

$$\mathcal{M}$$ is a sublinear operator, i.e., it satisfies $$\mathcal{M}(f + g) \leq \mathcal{M}(f) + \mathcal{M}(g)$$ and $$\mathcal{M}(\lambda f) = \vert \lambda \vert \mathcal{M}(f)$$ for all locally integrable functions f and g and all complex constants $$\lambda$$. It also has some interesting properties:

If f is locally integrable, then by considering the average of f over the ball B(x, | x | + R), which contains the ball B(0, R), we obtain

(2.1.1)

for all $$x \in \mathbf{R}^{n}$$, where v n is the volume of the unit ball in $$\mathbf{R}^{n}$$. An interesting consequence of (2.1.1) is the following: suppose that $$f\neq 0$$ on a set of positive measure E, then $$\mathcal{M}(f)$$ is not in $$L^{1}(\mathbf{R}^{n})$$. In other words, if f is in $$L_{\text{loc}}^{1}(\mathbf{R}^{n})$$ and $$\mathcal{M}(f)$$ is in $$L^{1}(\mathbf{R}^{n})$$, then f = 0 a.e. To see this, integrate (2.1.1) over the ball $$\mathbf{R}^{n}$$ to deduce that $$\|f\chi _{B(0,R)}\|_{L^{1}} = 0$$ and thus f(x) = 0 for almost all x in the ball B(0, R). Since this is valid for all $$R = 1,2,3,\ldots$$, it follows that f = 0 a.e. in $$\mathbf{R}^{n}$$.

Another remarkable locality property of $$\mathcal{M}$$ is that if $$\mathcal{M}(f)(x_{0}) = 0$$ for some x ₀ in $$\mathbf{R}^{n}$$, then f = 0 a.e. To see we take x = x ₀ in (2.1.1) to deduce that $$\|f\chi _{B(0,R)}\|_{L^{1}} = 0$$ and as before we have that f = 0 a.e. on every ball centered at the origin, i.e., f = 0 a.e. in $$\mathbf{R}^{n}$$.

A related analogue of $$\mathcal{M}(f)$$ is its uncentered version M(f), defined as the supremum of all averages of f over all open balls containing a given point.

Definition 2.1.3.

The uncentered Hardy–Littlewood maximal function of f,

is defined as the supremum of the averages of | f | over all open balls $$B(y,\delta )$$ that contain the point x.

Clearly $$\mathcal{M}(f) \leq M(f)$$; in other words, M is a larger operator than $$\mathcal{M}$$. However, $$M(f) \leq 2^{n}\mathcal{M}(f)$$ and the boundedness properties of M are identical to those of $$\mathcal{M}$$.

Example 2.1.4.

On $$\mathbf{R}$$, let f be the characteristic function of the interval I = [a, b]. For x ∈ (a, b), clearly M(f)(x) = 1. For x > b, a calculation shows that the largest average of f over all intervals $$(y-\delta,y+\delta )$$ that contain x is obtained when $$\delta = \frac{1} {2}(x - a)$$ and $$y = \frac{1} {2}(x + a)$$. Similarly, when x < a, the largest average is obtained when $$\delta = \frac{1} {2}(b - x)$$ and $$y = \frac{1} {2}(b + x)$$. We conclude that

Observe that M does not have a jump at x = a and x = b and is in fact equal to the function $$\big(1 + \frac{\mbox{ dist }\!(x,I)} {\vert I\vert } \big)^{-1}$$.

We are now ready to obtain some basic properties of maximal functions. We need the following simple covering lemma.

Lemma 2.1.5.

Let $$\{B_{1},B_{2},\ldots,B_{k}\}$$ be a finite collection of open balls in $$\mathbf{R}^{n}$$ . Then there exists a finite subcollection $$\{B_{j_{1}},\ldots,B_{j_{l}}\}$$ of pairwise disjoint balls such that

(2.1.2)

Proof.

Let us reindex the balls so that

Let j ₁ = 1. Having chosen $$j_{1},j_{2},\ldots,j_{i}$$, let j i+1 be the least index s > j i such that $$\bigcup _{m=1}^{i}B_{j_{m}}$$ is disjoint from B s . Since we have a finite number of balls, this process will terminate, say after l steps. We have now selected pairwise disjoint balls $$B_{j_{1}},\ldots,B_{j_{l}}$$. If some B m was not selected, that is, $$m\notin \{j_{1},\ldots,j_{l}\}$$, then B m must intersect a selected ball $$B_{j_{r}}$$ for some j r  < m. Then B m has smaller size than $$B_{j_{r}}$$ and we must have $$B_{m} \subseteq 3B_{j_{r}}$$. This shows that the union of the unselected balls is contained in the union of the triples of the selected balls. Therefore, the union of all balls is contained in the union of the triples of the selected balls. Thus

and the required conclusion follows. □ 

It was noted earlier that $$\mathcal{M}(f)$$ and M(f) never map into L ¹. However, it is true that these functions are in $$L^{1,\infty }$$ when f is in L ¹. Operators that map L ¹ to $$L^{1,\infty }$$ are said to be weak type (1, 1). The centered and uncentered maximal functions $$\mathcal{M}$$ and M are of weak type (1, 1) as shown in the next theorem.

Theorem 2.1.6.

​​ The uncentered and centered Hardy–Littlewood maximal operators M and $$\mathcal{M}$$ map $$L^{1}(\mathbf{R}^{n})$$ to $$L^{1,\infty }(\mathbf{R}^{n})$$ with constant at most 3 n and also $$L^{p}(\mathbf{R}^{n})$$ to $$L^{p}(\mathbf{R}^{n})$$ for $$1 < p < \infty $$ with constant at most $$3^{n/p}p(p - 1)^{-1}$$ . For any $$f \in L^{1}(\mathbf{R}^{n})$$ we also have

(2.1.3)

Proof.

We claim that the set $$E_{\alpha } =\{ x \in \mathbf{R}^{n}:\,\, M(f)(x) >\alpha \}$$ is open. Indeed, for $$x \in E_{\alpha }$$, there is an open ball B x that contains x such that the average of | f | over B x is strictly bigger than $$\alpha$$. Then the uncentered maximal function of any other point in B x is also bigger than $$\alpha$$, and thus $$B_{x}$$ is contained in $$E_{\alpha }$$. This proves that $$E_{\alpha }$$ is open.

Let K be a compact subset of $$E_{\alpha }$$. For each x ∈ K there exists an open ball B x containing the point x such that

(2.1.4)

Observe that $$B_{x} \subset E_{\alpha }$$ for all x. By compactness there exists a finite subcover $$\{B_{x_{1}},\ldots,B_{x_{k}}\}$$ of K. Using Lemma 2.1.5 we find a subcollection of pairwise disjoint balls $$B_{x_{j_{ 1}}},\ldots,B_{x_{j_{ l}}}$$ such that (2.1.2) holds. Using (2.1.4) and (2.1.2) we obtain

since all the balls $$B_{x_{j_{ i}}}$$ are disjoint and contained in $$E_{\alpha }$$. Taking the supremum over all compact $$K \subseteq E_{\alpha }$$ and using the inner regularity of Lebesgue measure, we deduce (2.1.3). We have now proved that M maps $$L^{1} \rightarrow L^{1,\infty }$$ with constant 3 n . It is a trivial fact that M maps $$L^{\infty }\rightarrow L^{\infty }$$ with constant 1. Since M is well defined and finite a.e. on $$L^{1} + L^{\infty }$$, it is also on $$L^{p}(\mathbf{R}^{n})$$ for $$1 < p < \infty $$. The Marcinkiewicz interpolation theorem (Theorem 1.​3.​2) implies that M maps $$L^{p}(\mathbf{R}^{n})$$ to $$L^{p}(\mathbf{R}^{n})$$ for all $$1 < p < \infty $$. Using Exercise 1.3.3, we obtain the following estimate for the operator norm of M on $$L^{p}(\mathbf{R}^{n})$$:

(2.1.5)

Observe that a direct application of Theorem 1.​3.​2 would give the slightly worse bound of $$2\big( \frac{p} {p-1}\big)^{\frac{1} {p} }3^{\frac{n} {p} }.$$ Finally the boundedness of $$\mathcal{M}$$ follows from that of M. □ 

Remark 2.1.7.

The previous proof gives a bound on the operator norm of M on $$L^{p}(\mathbf{R}^{n})$$ that grows exponentially with the dimension. One may wonder whether this bound could be improved to a better one that does not grow exponentially in the dimension n, as $$n \rightarrow \infty $$. This is not possible; see Exercise 2.1.8.

Example 2.1.8.

Let R > 0. Then we have

(2.1.6)

The lower estimate in (2.1.6), is an easy consequence of the fact that the ball $$B(x,\vert x\vert + R)$$ contains the ball B(0, R). For the upper estimate, we first consider the case where | x | ≤ 2R, when clearly $$M(\chi _{B(0,R)})(x) \leq 1 \leq \frac{3^{n}\,R^{n}} {(\vert x\vert +R)^{n}}$$. In the case where | x |  > 2R, if the balls B(x, r) and B(0, R) intersect, we must have that r >  | x | − R. But note that $$\vert x\vert - R > \frac{1} {3}(\vert x\vert + R)$$, since | x |  > 2R. We conclude that for | x |  > 2R we have

and thus the upper estimate in (2.1.6) holds since $$M(\chi _{B(0,R)}) \leq 2^{n}\mathcal{M}(\chi _{B(0,R)})$$. Thus in both cases the upper estimate in (2.1.6) is valid.

Next we estimate $$M(M(\chi _{B(0,R)}))(x)$$. First we write

Using the upper estimate in (2.1.6) and the sublinearity of M, we obtain

where the last estimate follows by summing separately over k satisfying $$2^{k+1} \leq \vert x\vert /R$$ and 2 k+1 ≥ | x | ∕R. Note that the presence of the logarithm does not affect the $$L^{p}$$ boundedness of this function when p > 1.

2.1.2 Control of Other Maximal Operators

We now study some properties of the Hardy–Littlewood maximal function. We begin with a notational definition that we plan to use throughout this book.

Definition 2.1.9.

Given a function g on $$\mathbf{R}^{n}$$ and $$\varepsilon > 0$$, we denote by $$g_{\varepsilon }$$ the following function:

(2.1.7)

As observed in Example 1.​2.​17, if g is an integrable function with integral equal to 1, then the family defined by (2.1.7) is an approximate identity. Therefore, convolution with $$g_{\varepsilon }$$ is an averaging operation. The Hardy–Littlewood maximal function $$\mathcal{M}(f)$$ is obtained as the supremum of the averages of a function f with respect to the dilates of the kernel $$k = v_{n}^{-1}\chi _{B(0,1)}$$ in $$\mathbf{R}^{n}$$; here v n is the volume of the unit ball B(0, 1). Indeed, we have

Note that the function $$k = v_{n}^{-1}\chi _{B(0,1)}$$ has integral equal to 1, and convolving with $$k_{\varepsilon }$$ is an averaging operation.

It turns out that the Hardy–Littlewood maximal function controls the averages of a function with respect to any radially decreasing L ¹ function. Recall that a function f on $$\mathbf{R}^{n}$$ is called radial if f(x) = f(y) whenever | x |  =  | y | . Note that a radial function f on $$\mathbf{R}^{n}$$ has the form $$f(x) =\varphi (\vert x\vert )$$ for some function $$\varphi$$ on $$\mathbf{R}^{+}$$. We have the following result.

Theorem 2.1.10.

Let k ≥ 0 be a function on $$[0,\infty )$$ that is continuous except at a finite number of points. Suppose that K(x) = k(|x|) is an integrable function on $$\mathbf{R}^{n}$$ that satisfies

(2.1.8)

i.e., k is decreasing. Then the following estimate is true:

(2.1.9)

for all locally integrable functions f on $$\mathbf{R}^{n}$$ .

Proof.

We prove (2.1.9) when K is radial, satisfies (2.1.8), and is compactly supported and continuous. When this case is established, select a sequence K j of radial, compactly supported, continuous functions that increase to K as $$j \rightarrow \infty $$. This is possible, since the function k is continuous except at a finite number of points. If (2.1.9) holds for each K j , passing to the limit implies that (2.1.9) also holds for K. Next, we observe that it suffices to prove (2.1.9) for x = 0. When this case is established, replacing f(t) by f(t + x) implies that (2.1.9) holds for all x.

Let us now fix a radial, continuous, and compactly supported function K with support in the ball B(0, R), satisfying (2.1.8). Also fix an $$f \in L_{\text{loc}}^{1}$$ and take x = 0. Let e ₁ be the vector $$(1,0,0,\ldots,0)$$ on the unit sphere $$\mathbf{S}^{n-1}$$. Polar coordinates give

(2.1.10)

Define functions

where d θ denotes surface measure on $$\mathbf{S}^{n-1}$$. Using these functions, (2.1.10), and integration by parts, we obtain

(2.1.11)

where two of the integrals are of Lebesgue–Stieltjes type and we used our assumptions that G(0) = 0, $$K_{\varepsilon }(0) < \infty $$, $$G(\varepsilon R) < \infty $$, and $$K_{\varepsilon }(\varepsilon Re_{1}) = 0$$. Let v n be the volume of the unit ball in $$\mathbf{R}^{n}$$. Since

it follows that the expression in (2.1.11) is dominated by

Here we used integration by parts and the fact that the surface measure of the unit sphere $$\mathbf{S}^{n-1}$$ is equal to nv n . See Appendix A.3. The theorem is now proved. □ 

Remark 2.1.11.

Theorem 2.1.10 can be generalized as follows. If K is an L ¹ function on $$\mathbf{R}^{n}$$ such that $$\vert K(x)\vert \leq k_{0}(\vert x\vert ) = K_{0}(x)$$, where k ₀ is a nonnegative decreasing function on $$[0,\infty )$$ that is continuous except at a finite number of points, then (2.1.9) holds with $$\|K\|_{L^{1}}$$ replaced by $$\|K_{0}\|_{L^{1}}$$. Such a K ₀ is called a radial decreasing majorant of K. This observation is formulated as the following corollary.

Corollary 2.1.12.

If a function $$\varphi$$ has an integrable radially decreasing majorant Φ, then the estimate

is valid for all locally integrable functions f on $$\mathbf{R}^{n}$$ .

Example 2.1.13.

Let

where c n is a constant such that

The function P is called the Poisson kernel . We define L ¹ dilates P t of the Poisson kernel P by setting

for t > 0. It is straightforward to verify that when n ≥ 2,

that is, $$P_{t}(x_{1},\ldots,x_{n})$$ is a harmonic function of the variables $$(x_{1},\ldots,x_{n},t)$$. Therefore, for $$f \in L^{p}(\mathbf{R}^{n})$$, $$1 \leq p < \infty $$, the function

is harmonic in $$\mathbf{R}_{+}^{n+1}$$ and converges to f(x) in L p (dx) as t → 0, since $$\{P_{t}\}_{t>0}$$ is an approximate identity. If we knew that f ∗ P t converged to f a.e. as $$t \rightarrow 0$$, then we could say that u(x, t) solves the Dirichlet problem

(2.1.12)

Solving the Dirichlet problem (2.1.12) motivates the study of the almost everywhere convergence of the expressions f ∗ P t .

Let us now compute the value of the constant c n . Denote by $$\omega _{n-1}$$ the surface area of $$\mathbf{S}^{n-1}$$. Using polar coordinates, we obtain

where we used the formula for $$\omega _{n-1}$$ in Appendix A.3 and an identity in Appendix A.4. We conclude that

and that the Poisson kernel on $$\mathbf{R}^{n}$$ is given by

(2.1.13)

Theorem 2.1.10 implies that the solution of the Dirichlet problem (2.1.12) is pointwise bounded by the Hardy–Littlewood maximal function of f.

2.1.3 Applications to Differentiation Theory

We continue this section by obtaining some applications of the boundedness of the Hardy–Littlewood maximal function in differentiation theory.

We now show that the weak type (1, 1) property of the Hardy–Littlewood maximal function implies almost everywhere convergence for a variety of families of functions. We deduce this from the more general fact that a certain weak type property for the supremum of a family of linear operators implies almost everywhere convergence.

Here is our setup. Let (X, μ), (Y, ν) be measure spaces and let $$0 < p \leq \infty $$, $$0 < q < \infty $$. Suppose that D is a dense subspace of L p (X, μ). This means that for all f ∈ L p and all $$\delta > 0$$ there exists a g ∈ D such that $$\|f - g\|_{L^{p}} <\delta$$. Suppose that for every $$\varepsilon > 0$$, $$T_{\varepsilon }$$ is a linear operator that maps $$L^{p}(X,\mu )$$ into a subspace of measurable functions, which are defined everywhere on Y. For y ∈ Y, define a sublinear operator

(2.1.14)

and assume that T ∗(f) is ν- measurable for any $$f \in L^{p}(X,\mu )$$. We have the following.

Theorem 2.1.14.

Let 0 < p < ∞, $$0 < q < \infty $$ , and $$T_{\varepsilon }$$ and T ∗ as previously. Suppose that for some B > 0 and all f ∈ L p (X) we have

(2.1.15)

and that for all f ∈ D,

(2.1.16)

exists and is finite ν-a.e. (and defines a linear operator on D). Then for all functions f in L p (X,μ) the limit (2.1.16) exists and is finite ν-a.e., and defines a linear operator T on L p (X) (uniquely extending T defined on D) that satisfies

(2.1.17)

for all functions f in L p (X).

Proof.

Given f in $$L^{p}$$, we define the oscillation of f:

We would like to show that for all $$f \in L^{p}$$ and $$\delta > 0$$,

(2.1.18)

Once (2.1.18) is established, given $$f \in L^{p}(X)$$, we obtain that $$O_{f}(y) = 0$$ for ν-almost all y, which implies that $$T_{\varepsilon }(f)(y)$$ is Cauchy for ν-almost all y, and it therefore converges ν-a.e. to some T(f)(y) as $$\varepsilon \rightarrow 0$$. The operator T defined this way on L p (X) is linear and extends T defined on D.

To approximate O f we use density. Given η > 0, find a function $$g \in D$$ such that $$\|f - g\|_{L^{p}} <\eta$$. Since $$T_{\varepsilon }(g) \rightarrow T(g)$$ ν-a.e, it follows that O g  = 0 ν-a.e. Using this fact and the linearity of the $$T_{\varepsilon }$$’s, we conclude that

Now for any δ > 0 we have

Letting $$\eta \rightarrow 0$$, we deduce (2.1.18). We conclude that $$T_{\varepsilon }(f)$$ is a Cauchy sequence, and hence it converges ν-a.e. to some T(f). Since $$\vert T(f)\vert \leq \vert T_{{\ast}}(f)\vert,$$ the conclusion (2.1.17) of the theorem follows easily. □ 

We now derive some applications. First we return to the issue of almost everywhere convergence of the expressions f ∗ P y , where P is the Poisson kernel.

Example 2.1.15.

Fix $$1 \leq p < \infty $$ and $$f \in L^{p}(\mathbf{R}^{n})$$. Let

be the Poisson kernel on $$\mathbf{R}^{n}$$ and let $$P_{\varepsilon }(x) =\varepsilon ^{-n}P(\varepsilon ^{-1}x)$$. We deduce from the previous theorem that the family $$f {\ast} P_{\varepsilon }$$ converges to f a.e. Let D be the set of all continuous functions with compact support on $$\mathbf{R}^{n}$$. Since the family $$(P_{\varepsilon })_{\varepsilon >0}$$ is an approximate identity, Theorem 1.​2.​19 (2) implies that for f in $$D$$ we have that $$f {\ast} P_{\varepsilon } \rightarrow f$$ uniformly on compact subsets of $$\mathbf{R}^{n}$$ and hence pointwise everywhere. In view of Theorem 2.1.10, the supremum of the family of linear operators $$T_{\varepsilon }(f) = f {\ast} P_{\varepsilon }$$ is controlled by the Hardy–Littlewood maximal function, and thus it maps L p to $$L^{p,\infty }$$ for $$1 \leq p < \infty $$. Theorem 2.1.14 now gives that $$f {\ast} P_{\varepsilon }$$ converges to f a.e. for all $$f \in L^{p}$$.

Here is another application of Theorem 2.1.14. Exercise 2.1.10 contains other applications.

Corollary 2.1.16.

(Lebesgue’s differentiation theorem) For any locally integrable function f on $$\mathbf{R}^{n}$$ we have

(2.1.19)

for almost all x in $$\mathbf{R}^{n}$$ . Consequently we have $$\vert f\vert \leq \mathcal{M}(f)$$ a.e. There is also an analogous statement to(2.1.19) in which balls are replaced by cubes centered at x. Precisely, for any locally integrable function f on $$\mathbf{R}^{n}$$ we have

(2.1.20)

for almost all x in $$\mathbf{R}^{n}$$ .

Proof.

Since $$\mathbf{R}^{n}$$ is the union of the balls B(0, N) for $$N = 1,2,3\ldots$$, it suffices to prove the required conclusion for almost all x inside a fixed ball B(0, N). Given a locally integrable function $$f$$ on $$\mathbf{R}^{n}$$, consider the function $$f_{N} = f\chi _{B(0,N+1)}$$. Then f N lies in $$L^{1}(\mathbf{R}^{n})$$. Let $$T_{\varepsilon }$$ be the operator given with convolution with $$k_{\varepsilon }$$, where $$k = v_{n}^{-1}\chi _{B(0,1)}$$ and $$0 <\varepsilon < 1$$. We know that the corresponding maximal operator T ∗ is controlled by the centered Hardy–Littlewood maximal function $$\mathcal{M}$$, which maps L ¹ to $$L^{1,\infty }$$. It is straightforward to verify that (2.1.19) holds for all continuous functions f with compact support. Since this set of functions is dense in L ¹, and T ∗ maps L ¹ to $$L^{1,\infty }$$, Theorem 2.1.14 implies that (2.1.19) holds for all integrable functions on $$\mathbf{R}^{n}$$, in particular for $$f_{N}$$. But for $$0 <\varepsilon < 1$$ and x ∈ B(0, N) we have $$f\chi _{B(x,\varepsilon )} = f_{N}\chi _{B(x,\varepsilon )}$$, so it follows that

for almost all $$x \in \mathbf{R}^{n}$$, in particular for almost all x in B(0, N). But on this set f N  = f, so the required conclusion follows. The assertion that $$\vert f\vert \leq \mathcal{M}(f)$$ a.e. is an easy consequence of (2.1.19) when the limit is replaced by a supremum.

Finally, with minor modifications, the proof can be adjusted to work for cubes in place of balls. To prove (2.1.20), for $$f \in L_{\mbox{ loc}}^{1}(\mathbf{R}^{n})$$ we introduce the maximal operator

Then Exercise 2.1.3 yields that $$\mathcal{M}_{c}$$ maps $$L^{1}(\mathbf{R}^{n})$$ to weak $$L^{1}(\mathbf{R}^{n})$$ and the preceding proof with $$\mathcal{M}_{c}$$ in place of $$\mathcal{M}$$ yields (2.1.20). □ 

The following corollaries were inspired by Example 2.1.15.

Corollary 2.1.17.

​​​ (Differentiation theorem for approximate identities) Let K be an L ¹ function on $$\mathbf{R}^{n}$$ with integral 1 that has a continuous integrable radially decreasing majorant. Then $$f {\ast} K_{\varepsilon } \rightarrow f$$ a.e. as $$\varepsilon \rightarrow 0$$ for all $$f \in L^{p}(\mathbf{R}^{n})$$ , $$1 \leq p < \infty $$ .

Proof.

It follows from Example 1.​2.​17 that $$K_{\varepsilon }$$ is an approximate identity. Theorem 1.​2.​19 now implies that $$f {\ast} K_{\varepsilon } \rightarrow f$$ uniformly on compact sets when f is continuous. Let D be the space of all continuous functions with compact support. Then $$f {\ast} K_{\varepsilon } \rightarrow f$$ a.e. for f ∈ D. It follows from Corollary 2.1.12 that $$T_{{\ast}}(f) =\sup _{\varepsilon >0}\vert f {\ast} K_{\varepsilon }\vert $$ maps L p to $$L^{p,\infty }$$ for $$1 \leq p < \infty $$. Using Theorem 2.1.14, we conclude the proof of the corollary. □ 

Remark 2.1.18.

Fix $$f \in L^{p}(\mathbf{R}^{n})$$ for some $$1 \leq p < \infty $$. Theorem 1.​2.​19 implies that $$f {\ast} K_{\varepsilon }$$ converges to f in $$L^{p}$$ and hence some subsequence $$f {\ast} K_{\varepsilon _{n}}$$ of $$f {\ast} K_{\varepsilon }$$ converges to f a.e. as $$n \rightarrow \infty $$, ($$\varepsilon _{n} \rightarrow 0$$). Compare this result with Corollary 2.1.17, which gives a.e. convergence for the whole family $$f {\ast} K_{\varepsilon }$$ as $$\varepsilon \rightarrow 0$$.

Corollary 2.1.19.

(Differentiation theorem for multiples of approximate identities) Let K be a function on $$\mathbf{R}^{n}$$ that has an integrable radially decreasing majorant. Let $$a =\int _{\mathbf{R}^{n}}K(x)\,dx$$ . Then for all $$f \in L^{p}(\mathbf{R}^{n})$$ and $$1 \leq p < \infty $$ , $$(f {\ast} K_{\varepsilon })(x) \rightarrow af(x)$$ for almost all $$x \in \mathbf{R}^{n}$$ as $$\varepsilon \rightarrow 0$$ .

Proof.

Use Theorem 1.​2.​21 instead of Theorem 1.​2.​19 in the proof of Corollary 2.1.17. □ 

The following application of the Lebesgue differentiation theorem uses a simple stopping-time argument. This is the sort of argument in which a selection procedure stops when it is exhausted at a certain scale and is then repeated at the next scale. A certain refinement of the following proposition is of fundamental importance in the study of singular integrals given in Chapter 4.​

Proposition 2.1.20.

Given a nonnegative integrable function f on $$\mathbf{R}^{n}$$ and $$\alpha > 0$$ , there exists a collection of disjoint (possibly empty) open cubes Q j such that for almost all $$x \in \big (\bigcup _{j}Q_{j}\big)^{c}$$ we have $$f(x) \leq \alpha$$ and

(2.1.21)

Proof.

The proof provides an excellent paradigm of a stopping-time argument. Start by decomposing $$\mathbf{R}^{n}$$ as a union of cubes of equal size, whose interiors are disjoint, and whose diameter is so large that $$\vert Q\vert ^{-1}\int _{Q}f(x)\,dx \leq \alpha$$ for every Q in this mesh. This is possible since f is integrable and $$\vert Q\vert ^{-1}\int _{Q}f(x)\,dx \rightarrow 0$$ as $$\vert Q\vert \rightarrow \infty $$. Call the union of these cubes $$\mathcal{E}_{0}$$.

Divide each cube in the mesh into 2 n congruent cubes by bisecting each of the sides. Call the new collection of cubes $$\mathcal{E}_{1}$$. Select a cube Q in $$\mathcal{E}_{1}$$ if

(2.1.22)

and call the set of all selected cubes $$\mathcal{S}_{1}$$. Now subdivide each cube in $$\mathcal{E}_{1}\setminus \mathcal{S}_{1}$$ into 2 n congruent cubes by bisecting each of the sides as before. Call this new collection of cubes $$\mathcal{E}_{2}$$. Repeat the same procedure and select a family of cubes $$\mathcal{S}_{2}$$ that satisfy (2.1.22). Continue this way ad infinitum and call the cubes in $$\bigcup _{m=1}^{\infty }\mathcal{S}_{m}$$ “selected.” If Q was selected, then there exists Q ₁ in $$\mathcal{E}_{m-1}$$ containing Q that was not selected at the (m − 1)th step for some m ≥ 1. Therefore,

Now call F the closure of the complement of the union of all selected cubes. If x ∈ F, then there exists a sequence of cubes containing x whose diameter shrinks down to zero such that the average of f over these cubes is less than or equal to $$\alpha$$. By Corollary 2.1.16, it follows that $$f(x) \leq \alpha$$ almost everywhere in F. This proves the proposition. □ 

In the proof of Proposition 2.1.20 it was not crucial to assume that f was defined on all $$\mathbf{R}^{n}$$, but only on a cube. We now give a local version of this result.

Corollary 2.1.21.

Let f ≥ 0 be an integrable function over a cube Q in $$\mathbf{R}^{n}$$ and let $$\alpha \geq \frac{1} {\vert Q\vert }\int _{Q}f\,dx$$ . Then there exist disjoint (possibly empty) open subcubes Q j of Q such that for almost all $$x \in Q\setminus \bigcup _{j}Q_{j}$$ we have $$f \leq \alpha$$ and(2.1.21) holds for all Q j .

Proof.

The proof easily follows by a simple modification of Proposition 2.1.20 in which $$\mathbf{R}^{n}$$ is replaced by the fixed cube Q. To apply Corollary 2.1.16, we extend f to be zero outside the cube Q. □ 

See Exercise 2.1.4 for an application of Proposition 2.1.20 involving maximal functions.

Exercises

2.1.1. A positive Borel measure $$\mu$$ on $$\mathbf{R}^{n}$$ is called inner regular if for any open subset U of $$\mathbf{R}^{n}$$ we have $$\mu (U) =\sup \{\mu (K):\,\, K \subseteq U,\,\,\,\mbox{ $K$ compact}\}$$ and μ is called locally finite if $$\mu (B) < \infty $$ for all balls B. (a) Let μ be a positive inner regular locally finite measure on $$\mathbf{R}^{n}$$ that satisfies the following doubling condition: There exists a constant D(μ) > 0 such that for all $$x \in \mathbf{R}^{n}$$ and r > 0 we have

For $$f \in L_{\mbox{ loc}}^{1}(\mathbf{R}^{n},\mu )$$ define the uncentered maximal function $$M_{\mu }(f)$$ with respect to μ by

Show that M μ maps $$L^{1}(\mathbf{R}^{n},\mu )$$ to $$L^{1,\infty }(\mathbf{R}^{n},\mu )$$ with constant at most D(μ) and $$L^{p}(\mathbf{R}^{n},\mu )$$ to itself with constant at most $$2\big( \frac{p} {p-1}\big)^{\frac{1} {p} }D(\mu )^{\frac{1} {p} }$$.

(b) Obtain as a consequence a differentiation theorem analogous to Corollary 2.1.16.

[Hint: Part (a): For $$f \in L^{1}(\mathbf{R}^{n},\mu )$$ show that the set $$E_{\alpha } =\{ M_{\mu }(f) >\alpha \}$$ is open. Then use the argument of the proof of Theorem 2.1.6 and the inner regularity of μ.]

2.1.2.

On $$\mathbf{R}$$ consider the maximal function M μ of Exercise 2.1.1.

(a) (W. H. Young) Prove the following covering lemma. Given a finite set $$\mathcal{F}$$ of open intervals in $$\mathbf{R}$$, prove that there exist two subfamilies each consisting of pairwise disjoint intervals such that the union of the intervals in the original family is equal to the union of the intervals of both subfamilies. Use this result to show that the maximal function M μ of Exercise 2.1.1 maps $$L^{1}(\mu ) \rightarrow L^{1,\infty }(\mu )$$ with constant at most 2.

(b) ([134]) Prove that for any $$\sigma$$-finite positive measure μ on $$\mathbf{R}$$, $$\alpha > 0$$, and $$f \in L_{\text{loc}}^{1}(\mathbf{R},\mu )$$ we have

Use this result and part (a) to prove that for all $$\alpha > 0$$ and all locally integrable f we have

and note that equality is obtained when α = 1 and $$f(x) = \vert x\vert ^{-1/p}$$.

(c) Conclude that $$M_{\mu }$$ maps L p (μ) to L p (μ), $$1 < p < \infty $$, with bound at most the unique positive solution A p of the equation

(d) ([136]) If μ is the Lebesgue measure show that for $$1 < p < \infty $$ we have

where A p is the unique positive solution of the equation in part (c).

[Hint: Part (a): Select a subset $$\mathcal{G}$$ of $$\mathcal{F}$$ with minimal cardinality such that $$\bigcup _{J\in \mathcal{G}}J =\bigcup _{I\in \mathcal{F}}I$$. Part (d): One direction follows from part (c). Conversely, $$M(\vert x\vert ^{-1/p})(1) = \frac{p} {p-1} \frac{\gamma ^{1/p'}+1} {\gamma +1},$$ where $$\gamma$$ is the unique positive solution of the equation $$\frac{p} {p-1} \frac{\gamma ^{1/p'}+1} {\gamma +1} =\gamma ^{-1/p}.$$ Conclude that $$M(\vert x\vert ^{-1/p})(1) = A_{p}$$ and that $$M(\vert x\vert ^{-1/p}) = A_{p}\vert x\vert ^{-1/p}$$. Since this function is not in $$L^{p}$$, consider the family $$f_{\varepsilon }(x) = \vert x\vert ^{-1/p}\min (\vert x\vert ^{-\varepsilon },\vert x\vert ^{\varepsilon }),$$ $$\varepsilon > 0$$, and show that $$M(f_{\varepsilon })(x) \geq (1 +\gamma ^{\frac{1} {p'}+\varepsilon })(1+\gamma )^{-1}(\frac{1} {p'}+\varepsilon )^{-1}f_{\varepsilon }(x)$$ for $$0 <\varepsilon < p'$$.]

2.1.3.

Define the centered Hardy–Littlewood maximal function $$\mathcal{M}_{c}\!\!\!\!$$ and the uncentered Hardy–Littlewood maximal function M c using cubes with sides parallel to the axes instead of balls in $$\mathbf{R}^{n}$$. Prove that

where $$v_{n}$$ is the volume of the unit ball in $$\mathbf{R}^{n}$$. Conclude that $$\mathcal{M}_{c}$$ and $$M_{c}$$ are weak type (1, 1) and they map $$L^{p}(\mathbf{R}^{n})$$ to $$L^{p}(\mathbf{R}^{n})$$ for $$1 < p \leq \infty $$.

2.1.4.

(a) Prove the estimate:

and conclude that M maps $$L^{p}$$ to $$L^{p,\infty }$$ with norm at most $$2 \cdot 3^{n/p}$$ for $$1 \leq p < \infty $$.

(b) Deduce that if $$f\,\log ^{+}(2\vert f\vert )$$ is integrable over a ball B, then M(f) is integrable over the same ball B.

(c) ([375], [336]) Apply Proposition 2.1.20 to | f | and α > 0 and Exercise 2.1.3 to show that with $$c_{n} = 2^{n}(n^{n/2}v_{n})^{-1}$$ we have

(d) Suppose that f is integrable and supported in a ball $$B(0,\rho )$$. Show that for x in $$B(0,2\rho )\setminus B(0,\rho )$$ we have $$\mathcal{M}(f)(x) \leq \mathcal{M}(f)(\rho ^{2}\vert x\vert ^{-2}x).$$ Conclude that

and from this deduce a similar inequality for M(f).

(e) Suppose that f is integrable and supported in a ball B and that M(f) is integrable over B. Let $$\lambda _{0} = 2^{n}\vert B\vert ^{-1}\|f\|_{L^{1}}$$. Use part (b) to prove that $$f\log ^{+}(\lambda _{0}^{-1}c_{n}\,\vert f\vert )$$ is integrable over B.

[Hint: Part (a): Write $$f = f\chi _{\vert f\vert >\alpha } + f\chi _{\vert f\vert \leq \alpha }$$. Part (b): Show that $$M(f\chi _{E})$$ is integrable over B, where E = { | f | ≥ 1∕2}. Part (c): Use Proposition 2.1.20. Part (d): Let $$x' =\rho ^{2}\vert x\vert ^{-2}x$$ for some $$\rho < \vert x\vert < 2\rho$$. Show that for R >  | x | −ρ, we have that

by showing that $$B(x,R) \cap B(0,\rho ) \subset B(x',R)$$. Part (e): For $$x\notin 2B$$ we have $$M(f)(x) \leq \lambda _{0}$$, hence $$\int _{2B}M(f)(x)\,dx \geq \int _{\lambda _{0}}^{\infty }\vert \{x \in 2B:\,\, M(f)(x) >\alpha \} \vert \,d\alpha$$.]

2.1.5.

(A. Kolmogorov) Let $$S$$ be a sublinear operator that maps $$L^{1}(\mathbf{R}^{n})$$ to $$L^{1,\infty }(\mathbf{R}^{n})$$ with norm B. Suppose that $$f \in L^{1}(\mathbf{R}^{n})$$. Prove that for any set A of finite Lebesgue measure and for all $$0 < q < 1$$ we have

and in particular, for the Hardy–Littlewood maximal operator,

[Hint: Use the identity

and estimate the last measure by $$\min (\vert A\vert, \frac{B} {\alpha } \|f\|_{L^{1}})$$.]

2.1.6.

Let M s (f)(x) be the supremum of the averages of | f | over all rectangles with sides parallel to the axes containing x. The operator M s is called the strong maximal function .

(a) Prove that M s maps $$L^{p}(\mathbf{R}^{n})$$ to itself.

(b) Show that the operator norm of M s is $$A_{p}^{n}$$, where A p is as in Exercise 2.1.2 (c).

(c) Prove that M s is not weak type (1,1).

2.1.7.

Prove that if

for some $$A,\varepsilon > 0$$, and $$\varphi _{t_{1},\ldots,t_{n}}(x) = t_{1}^{-1}\cdots t_{n}^{-1}\varphi (t_{1}^{-1}x_{1},\ldots,t_{n}^{-1}x_{n}),$$ then the maximal operator

is pointwise controlled by the strong maximal function.

2.1.8.

Prove that for any fixed $$1 < p < \infty $$, the operator norm of M on $$L^{p}(\mathbf{R}^{n})$$ tends to infinity as $$n \rightarrow \infty $$.

[Hint: Let f ₀ be the characteristic function of the unit ball in $$\mathbf{R}^{n}$$. Consider the averages $$\vert B_{x}\vert ^{-1}\int _{B_{x}}f_{0}\,dy$$, where $$B_{x} = B\big(\frac{1} {2}(\vert x\vert -\vert x\vert ^{-1}) \frac{x} {\vert x\vert }, \frac{1} {2}(\vert x\vert + \vert x\vert ^{-1})\big)$$ for | x |  > 1.]

2.1.9.

(a) In $$\mathbf{R}^{2}$$ let $$M_{0}(f)(x)$$ be the maximal function obtained by taking the supremum of the averages of | f | over all rectangles (of arbitrary orientation) containing x. Prove that M ₀ is not bounded on $$L^{p}(\mathbf{R}^{n})$$ for p ≤ 2 and conclude that M ₀ is not weak type $$(1,1)$$.

(b) Let $$M_{00}(f)(x)$$ be the maximal function obtained by taking the supremum of the averages of | f | over all rectangles in $$\mathbf{R}^{2}$$ of arbitrary orientation but fixed eccentricity containing x. (The eccentricity of a rectangle is the ratio of its longer side to its shorter side.) Using a covering lemma, show that M ₀₀ is weak type (1, 1) with a bound proportional to the square of the eccentricity.

(c) On R n define a maximal function by taking the supremum of the averages of | f | over all products of intervals $$I_{1} \times \ldots \times I_{n}$$ containing a point x with $$\vert I_{2}\vert = a_{2}\vert I_{1}\vert,\ldots,\vert I_{n}\vert = a_{n}\vert I_{1}\vert $$ and $$a_{2},\ldots,a_{n} > 0$$ fixed. Show that this maximal function is of weak type (1, 1) with bound independent of the numbers $$a_{2},\ldots,a_{n}$$.

[Hint: Part (b): Let b be the eccentricity. If two rectangles with the same eccentricity intersect, then the smaller one is contained in the bigger one scaled 4b times. Then use an argument similar to that in Lemma 2.1.5.]

2.1.10.

(a) Let 0 < p, q < ∞ and let X, Y be measure spaces. Suppose that $$T_{\varepsilon }$$ are maps from $$L^{p}(X)$$ to $$L^{q,\infty }(Y )$$ satisfy $$\vert T_{\varepsilon }(f + g)\vert \leq K(\vert T_{\varepsilon }(f)\vert + \vert T_{\varepsilon }(g)\vert )$$ for all $$\varepsilon > 0$$ and all $$f,g \in L^{p}(X)$$, and also $$\lim _{\varepsilon \rightarrow 0}T_{\varepsilon }(f) = 0$$ a.e. for all f in some dense subspace D of L p (X). Assume furthermore that the maximal operator $$T_{{\ast}}(f) =\sup _{\varepsilon >0}\vert T_{\varepsilon }(f)\vert $$ maps L p (X) to $$L^{q,\infty }(Y )$$. Prove that $$\lim _{\varepsilon \rightarrow 0}T_{\varepsilon }(f) = 0$$ a.e. for all f in L p (X).

(b) Use the result in part (a) to prove the following version of the Lebesgue differentiation theorem : Let $$f \in L^{p}(\mathbf{R}^{n})$$ for some $$0 < p < \infty $$. Then for almost all $$x \in \mathbf{R}^{n}$$ we have

where the limit is taken over all open balls B containing x and shrinking to {x}.

(c) Conclude that for any f in $$L_{\text{loc}}^{1}(\mathbf{R}^{n})$$ and for almost all $$x \in \mathbf{R}^{n}$$ we have

where the limit is taken over all open balls B containing x and shrinking to {x}.

[Hint: (a) Define an oscillation $$O_{f}(y) =\limsup _{\varepsilon \rightarrow 0}\vert T_{\varepsilon }(f)(y)\vert $$. For all f in L p (X) and g ∈ D we have that $$O_{f}(y) \leq KO_{f-g}(y)$$. Then use the argument in the proof of Theorem 2.1.14. (b) Apply part (a) with

observing that $$T_{{\ast}}(f) =\sup _{\varepsilon >0}T_{\varepsilon }(f) \leq \max (1,2^{\frac{1-p} {p} })\big(\vert f\vert + M(\vert f\vert ^{p})^{\frac{1} {p} }\big)$$. (c) Follows from part (b) with p = 1. Note that part (b) can be proved without part (a) but using part (c) as follows: for every rational number a there is a set E a of Lebesgue measure zero such that for $$x \in \mathbf{R}^{n}\setminus E_{a}$$ we have $$\lim _{B\ni x,\begin{array}{c}\vert B\vert \rightarrow 0\end{array}} \frac{1} {\vert B\vert }\int _{B}\vert g(y) - a\vert ^{p}\,dy = \vert g(x) - a\vert ^{p}$$, since the function $$y\mapsto \vert f(y) - a\vert ^{p}$$ is in $$L_{\text{loc}}^{1}(\mathbf{R}^{n})$$. By considering an enumeration of the rationals, find a set of measure zero E such for $$x\notin E$$ the preceding limit exists for all rationals a and by continuity for all real numbers a, in particular for a = g(x).]

2.1.11.

Let f be in $$L^{1}(\mathbf{R})$$. Define the right maximal function M R (f) and the left maximal function M L (f) as follows:

(a) Show that for all $$\alpha > 0$$ and $$f \in L^{1}(\mathbf{R})$$ we have

(b) Extend the definition of M L (f) and M R (f) for $$f \in L^{p}(\mathbf{R})$$ for $$1 \leq p \leq \infty $$. Show that M L and M R map $$L^{p}$$ to $$L^{p}$$ with norm at most p∕(p − 1) for all p with $$1 < p < \infty $$.

(c) Construct examples to show that the operator norms of M L and M R on $$L^{p}(\mathbf{R})$$ are exactly p∕(p − 1) for $$1 < p < \infty $$.

(d) Prove that $$M =\max (M_{R},M_{L})$$.

(e) Let $$N =\min (M_{R},M_{L})$$. Obtain the following consequence of part (a),

(f) Use part (e) to prove that

[Hint: (a) Write the set $$E_{\alpha } =\{ M_{R}(f) >\alpha \}$$ as a union of open intervals $$(a_{j},b_{j})$$. For each x in $$(a_{j},b_{j})$$, let $$N_{x} =\big\{ s \in \mathbf{R}:\,\,\int _{ x}^{s}\vert f\vert >\alpha (s - x)\big\} \cap (x,b_{j}]$$. Show that N x is nonempty and that $$\sup N_{x} = b_{j}$$ for every $$x \in (a_{j},b_{j})$$. Conclude that $$\int _{a_{j}}^{b_{j}}\vert f(t)\vert \,dt \geq \alpha (b_{j} - a_{j})$$, which implies that each $$a_{j}$$ is finite. For the reverse inequality use that $$a_{j}\notin E_{\alpha }$$. Part (d) is due to K. L. Phillips. (e) First obtain a version of the equality with M R in the place of M and M L in the place of N. Then use that $$M(f)^{q} + N(f)^{q} = M_{L}(f)^{q} + M_{R}(f)^{q}$$ for all q. (f) Use that $$\vert f\vert \,N(f)^{p-1} \leq \frac{1} {p}\vert f\vert ^{p} + \frac{1} {p'}N(f)^{p}$$. This alternative proof of the result in Exercise 2.1.2(c) was suggested by J. Duoandikoetxea.]

2.1.12. A cube $$Q = [a_{1}2^{k},(a_{1} + 1)2^{k}) \times \cdots \times [a_{n}2^{k},(a_{n} + 1)2^{k})$$ on $$\mathbf{R}^{n}$$ is called dyadic if k, $$a_{1},\ldots,a_{n} \in \mathbf{Z}$$. Observe that either two dyadic cubes are disjoint or one contains the other. Define the dyadic maximal function

where the supremum is taken over all dyadic cubes Q containing x. (a) Prove that M d maps L ¹ to $$L^{1,\infty }$$ with constant at most one. Presicely, show that for all $$\alpha > 0$$ and $$f \in L^{1}(\mathbf{R}^{n})$$ we have

(b) Conclude that M d maps $$L^{p}(\mathbf{R}^{n})$$ to itself with constant at most p∕(p − 1).

2.1.13. Observe that the proof of Theorem 2.1.6 yields the estimate

for $$\lambda > 0$$ and f locally integrable. Use the result of Exercise 1.1.12(a) to prove that the Hardy–Littlewood maximal operator M maps the space $$L^{p,\infty }(\mathbf{R}^{n})$$ to itself for $$1 < p < \infty $$.

2.1.14. Let $$K(x) = (1 + \vert x\vert )^{-n-\delta }$$ be defined on $$\mathbf{R}^{n}$$. Prove that there exists a constant $$C_{n,\delta }$$ such that for all $$\varepsilon _{0} > 0$$ we have the estimate

for all f locally integrable on $$\mathbf{R}^{n}$$.

[Hint: Apply only a minor modification to the proof of Theorem 2.1.10.]

2.2 The Schwartz Class and the Fourier Transform

In this section we introduce the single most important tool in harmonic analysis, the Fourier transform. It is often the case that the Fourier transform is introduced as an operation on L ¹ functions. In this exposition we first define the Fourier transform on a smaller class, the space of Schwartz functions, which turns out to be a very natural environment. Once the basic properties of the Fourier transform are derived, we extend its definition to other spaces of functions.

We begin with some preliminaries. Given $$x = (x_{1},\ldots,x_{n}) \in \mathbf{R}^{n}$$, we set $$\vert x\vert = (x_{1}^{2} +\ldots +x_{n}^{2})^{1/2}$$. The partial derivative of a function f on $$\mathbf{R}^{n}$$ with respect to the jth variable x j is denoted by $$\partial _{j}f$$ while the mth partial derivative with respect to the jth variable is denoted by $$\partial _{j}^{m}f$$. The gradient of a function f is the vector $$\nabla f = (\partial _{1}f,\ldots,\partial _{n}f)$$. A multi-index $$\alpha$$ is an ordered n-tuple of nonnegative integers. For a multi-index $$\alpha = (\alpha _{1},\ldots,\alpha _{n})$$, $$\partial ^{\alpha }f$$ denotes the derivative $$\partial _{1}^{\alpha _{1}}\cdots \partial _{n}^{\alpha _{n}}f$$. If $$\alpha = (\alpha _{1},\ldots,\alpha _{n})$$ is a multi-index, $$\vert \alpha \vert =\alpha _{1} +\ldots +\alpha _{n}$$ denotes its size and $$\alpha ! =\alpha _{1}!\cdots \alpha _{n}!$$ denotes the product of the factorials of its entries. The number $$\vert \alpha \vert $$ indicates the total order of differentiation of $$\partial ^{\alpha }f$$. The space of functions in $$\mathbf{R}^{n}$$ all of whose derivatives of order at most $$N \in \mathbf{Z}^{+}$$ are continuous is denoted by $$\mathcal{C}^{N}(\mathbf{R}^{n})$$ and the space of all infinitely differentiable functions on $$\mathbf{R}^{n}$$ by $$\mathcal{C}^{\infty }(\mathbf{R}^{n})$$. The space of $$\mathcal{C}^{\infty }$$ functions with compact support on $$\mathbf{R}^{n}$$ is denoted by $$\mathcal{C}_{0}^{\infty }(\mathbf{R}^{n})$$. This space is nonempty; see Exercise 2.2.1(a).

For $$x \in \mathbf{R}^{n}$$ and $$\alpha = (\alpha _{1},\ldots,\alpha _{n})$$ a multi-index, we set $$x^{\alpha } = x_{1}^{\alpha _{1}}\cdots x_{n}^{\alpha _{n}}$$. Multi-indices will be denoted by the letters $$\alpha,\beta,\gamma,\delta,...$$. It is a simple fact to verify that

(2.2.1)

for some constant that depends on the dimension n and on $$\alpha$$. In fact, $$c_{n,\alpha }$$ is the maximum of the continuous function $$(x_{1},\ldots,x_{n})\mapsto \vert x_{1}^{\alpha _{1}}\cdots x_{n}^{\alpha _{n}}\vert $$ on the sphere $$\mathbf{S}^{n-1} =\{ x \in \mathbf{R}^{n}:\,\, \vert x\vert = 1\}$$. The converse inequality in (2.2.1) fails. However, the following substitute of the converse of (2.2.1) is of great use: for $$k \in \mathbf{Z}^{+}$$ we have

(2.2.2)

for all $$x \in \mathbf{R}^{n}\setminus \{0\}$$. To prove (2.2.2), take 1∕C n, k to be the minimum of the function

on $$\mathbf{S}^{n-1}$$; this minimum is positive since this function has no zeros on $$\mathbf{S}^{n-1}$$. A related inequality is

(2.2.3)

This follows from (2.2.2) for | x | ≥ 1, while for | x |  < 1 we note that the sum in (2.2.3) is at least one since $$\vert x^{(0,\ldots,0)}\vert = 1$$.

We end the preliminaries by noting the validity of the one-dimensional Leibniz rule

(2.2.4)

for all $$\mathcal{C}^{m}$$ functions f, g on $$\mathbf{R}$$, and its multidimensional analogue

(2.2.5)

for f, g in $$\mathcal{C}^{\vert \alpha \vert }(\mathbf{R}^{n})$$ for some multi-index $$\alpha$$, where the notation $$\beta \leq \alpha$$ in (2.2.5) means that β ranges over all multi-indices satisfying $$0 \leq \beta _{j} \leq \alpha _{j}$$ for all $$1 \leq j \leq n$$. We observe that identity (2.2.5) is easily deduced by repeated application of (2.2.4), which in turn is obtained by induction.

2.2.1 The Class of Schwartz Functions

We now introduce the class of Schwartz functions on $$\mathbf{R}^{n}$$. Roughly speaking, a function is Schwartz if it is smooth and all of its derivatives decay faster than the reciprocal of any polynomial at infinity. More precisely, we give the following definition.

Definition 2.2.1.

A $$\mathcal{C}^{\infty }$$ complex-valued function f on $$\mathbf{R}^{n}$$ is called a Schwartz function if for every pair of multi-indices $$\alpha$$ and $$\beta$$ there exists a positive constant $$C_{\alpha,\beta }$$ such that

(2.2.6)

The quantities $$\rho _{\alpha,\beta }(f)$$ are called the Schwartz seminorms of f. The set of all Schwartz functions on $$\mathbf{R}^{n}$$ is denoted by $$\mathcal{S}(\mathbf{R}^{n})$$.

Example 2.2.2.

The function $$e^{-\vert x\vert ^{2} }$$ is in $$\mathcal{S}(\mathbf{R}^{n})$$ but $$e^{-\vert x\vert }$$ is not, since it fails to be differentiable at the origin. The $$\mathcal{C}^{\infty }$$ function $$g(x) = (1 + \vert x\vert ^{4})^{-a}$$, $$a > 0$$, is not in $$\mathcal{S}$$ since it decays only like the reciprocal of a fixed polynomial at infinity. The set of all smooth functions with compact support, $$\mathcal{C}_{0}^{\infty }(\mathbf{R}^{n})$$, is contained in $$\mathcal{S}(\mathbf{R}^{n})$$.

Remark 2.2.3.

If f ₁ is in $$\mathcal{S}(\mathbf{R}^{n})$$ and f ₂ is in $$\mathcal{S}(\mathbf{R}^{m})$$, then the function of m + n variables $$f_{1}(x_{1},\ldots,x_{n})f_{2}(x_{n+1},\ldots,x_{n+m})$$ is in $$\mathcal{S}(\mathbf{R}^{n+m})$$. If f is in $$\mathcal{S}(\mathbf{R}^{n})$$ and P(x) is a polynomial of n variables, then P(x)f(x) is also in $$\mathcal{S}(\mathbf{R}^{n})$$. If $$\alpha$$ is a multi-index and $$f$$ is in $$\mathcal{S}(\mathbf{R}^{n})$$, then $$\partial ^{\alpha }f$$ is in $$\mathcal{S}(\mathbf{R}^{n})$$. Also note that

Remark 2.2.4.

The following alternative characterization of Schwartz functions is very useful. A $$\mathcal{C}^{\infty }$$ function $$f$$ is in $$\mathcal{S}(\mathbf{R}^{n})$$ if and only if for all positive integers N and all multi-indices $$\alpha$$ there exists a positive constant $$C_{\alpha,N}$$ such that

(2.2.7)

The simple proofs are omitted. We now discuss convergence in $$\mathcal{S}(\mathbf{R}^{n})$$.

Definition 2.2.5.

Let f k , f be in $$\mathcal{S}(\mathbf{R}^{n})$$ for $$k = 1,2,\ldots$$. We say that the sequence f k converges to f in $$\mathcal{S}(\mathbf{R}^{n})$$ if for all multi-indices $$\alpha$$ and $$\beta$$ we have

For instance, for any fixed $$x_{0} \in \mathbf{R}^{n}$$, $$f(x + x_{0}/k) \rightarrow f(x)$$ in $$\mathcal{S}(\mathbf{R}^{n})$$ for any $$f$$ in $$\mathcal{S}(\mathbf{R}^{n})$$ as $$k \rightarrow \infty $$.

This notion of convergence is compatible with a topology on $$\mathcal{S}(\mathbf{R}^{n})$$ under which the operations $$(f,g)\mapsto f + g$$, (a, f) → af, and $$f\mapsto \partial ^{\alpha }f$$ are continuous for all complex scalars a and multi-indices $$\alpha$$ ($$f,g \in \mathcal{S}(\mathbf{R}^{n})$$). A subbasis for open sets containing 0 in this topology is

for all $$\alpha$$, $$\beta$$ multi-indices and all r ∈ Q ⁺. Observe the following: If $$\rho _{\alpha,\beta }(f) = 0$$, then $$f = 0$$. This means that $$\mathcal{S}(\mathbf{R}^{n})$$ is a locally convex topological vector space equipped with the family of seminorms $$\rho _{\alpha,\beta }$$ that separate points. We refer to Reed and Simon [286] for the pertinent definitions. Since the origin in $$\mathcal{S}(\mathbf{R}^{n})$$ has a countable base, this space is metrizable. In fact, the following is a metric on $$\mathcal{S}(\mathbf{R}^{n})$$:

where $$\rho _{j}$$ is an enumeration of all the seminorms $$\rho _{\alpha,\beta }$$, $$\alpha$$ and $$\beta$$ multi-indices. One may easily verify that $$\mathcal{S}$$ is complete with respect to the metric d. Indeed, a Cauchy sequence $$\{h_{j}\}_{j}$$ in $$\mathcal{S}$$ would have to be Cauchy in $$L^{\infty }$$ and therefore it would converge uniformly to some function h. The same is true for the sequences $$\{\partial ^{\beta }h_{j}\}_{j}$$ and $$\{x^{\alpha }h_{j}(x)\}_{j}$$, and the limits of these sequences can be shown to be the functions $$\partial ^{\beta }h$$ and $$x^{\alpha }h(x)$$, respectively. It follows that the sequence $$\{h_{j}\}$$ converges to h in $$\mathcal{S}$$. Therefore, $$\mathcal{S}(\mathbf{R}^{n})$$ is a Fréchet space (complete metrizable locally convex space).

We note that convergence in $$\mathcal{S}$$ is stronger than convergence in all $$L^{p}$$. We have the following.

Proposition 2.2.6.

Let f, f k , $$k = 1,2,3,\ldots$$ , be in $$\mathcal{S}(\mathbf{R}^{n})$$ . If $$f_{k} \rightarrow f$$ in $$\mathcal{S}$$ then $$f_{k} \rightarrow f$$ in $$L^{p}$$ for all $$0 < p \leq \infty $$ . Moreover, there exists a C p,n > 0 such that

(2.2.8)

for all f for which the right-hand side is finite.

Proof.

Observe that when $$p < \infty $$ we have

The preceding inequality is also trivially valid when $$p = \infty $$. Now set $$m =\big [\frac{n+1} {p} \big] + 1$$ and use (2.2.2) to obtain

Conclusion (2.2.8) now follows immediately. This shows that convergence in $$\mathcal{S}$$ implies convergence in $$L^{p}$$. □ 

We now show that the Schwartz class is closed under certain operations.

Proposition 2.2.7.

Let f, g be in $$\mathcal{S}(\mathbf{R}^{n})$$ . Then fg and f ∗ g are in $$\mathcal{S}(\mathbf{R}^{n})$$ . Moreover,

(2.2.9)

for all multi-indices $$\alpha$$ .

Proof.

Fix f and g in $$\mathcal{S}(\mathbf{R}^{n})$$. Let e j be the unit vector $$(0,\ldots,1,\ldots,0)$$ with 1 in the jth entry and zeros in all the other entries. Since

(2.2.10)

as h → 0, and since the expression in (2.2.10) is pointwise bounded by some constant depending on f, the integral of the expression in (2.2.10) with respect to the measure $$g(x - y)\,dy$$ converges to zero as h → 0 by the Lebesgue dominated convergence theorem. This proves (2.2.9) when $$\alpha = (0,\ldots,1,\ldots,0)$$. The general case follows by repeating the previous argument and using induction.

We now show that the convolution of two functions in $$\mathcal{S}$$ is also in $$\mathcal{S}$$. For each N > 0 there is a constant C N such that

(2.2.11)

Inserting the simple estimate

in (2.2.11) we obtain that

This shows that f ∗ g decays like (1 + | x | )−N at infinity, but since N > 0 is arbitrary it follows that f ∗ g decays faster than the reciprocal of any polynomial.

Since $$\partial ^{\alpha }(f {\ast} g) = (\partial ^{\alpha }f) {\ast} g$$, replacing f by $$\partial ^{\alpha }f$$ in the previous argument, we also conclude that all the derivatives of f ∗ g decay faster than the reciprocal of any polynomial at infinity. Using (2.2.7), we conclude that f ∗ g is in $$\mathcal{S}$$. Finally, the fact that fg is in $$\mathcal{S}$$ follows directly from Leibniz’s rule (2.2.5) and (2.2.7). □ 

2.2.2 The Fourier Transform of a Schwartz Function

The Fourier transform is often introduced as an operation on L ¹. In that setting, problems of convergence arise when certain manipulations of functions are performed. Also, Fourier inversion requires the additional assumption that the Fourier transform is in L ¹. Here we initially introduce the Fourier transform on the space of Schwartz functions. The rapid decay of Schwartz functions at infinity allows us to develop its fundamental properties without encountering any convergence problems. The Fourier transform is a homeomorphism of the Schwartz class and Fourier inversion holds in it. For these reasons, this class is a natural environment for it.

For $$x = (x_{1},\ldots,x_{n})$$, $$y = (y_{1},\ldots,y_{n})$$ in $$\mathbf{R}^{n}$$ we use the notation

Definition 2.2.8.

Given f in $$\mathcal{S}(\mathbf{R}^{n})$$ we define

We call $$\widehat{f}$$ the Fourier transform of f.

Example 2.2.9.

If $$f(x) = e^{-\pi \vert x\vert ^{2} }$$ defined on $$\mathbf{R}^{n}$$, then $$\widehat{f}(\xi ) = f(\xi )$$. To prove this, observe that the function

defined on the line is constant (and thus equal to $$\int _{-\infty }^{+\infty }e^{-\pi t^{2} }dt$$), since its derivative is

Using this fact, we calculate the Fourier transform of the function $$t\mapsto e^{-\pi t^{2} }$$ on $$\mathbf{R}$$ by completing the squares as follows:

where $$\tau \in \mathbf{R}$$, and we used that

(2.2.12)

a fact that can be found in Appendix A.1.

Remark 2.2.10.

It follows from the definition of the Fourier transform that if f is in $$\mathcal{S}(\mathbf{R}^{n})$$ and g is in $$\mathcal{S}(\mathbf{R}^{m})$$, then

where the first $$\,\widehat{\,\,}\,$$ denotes the Fourier transform on R n+m . In other words, the Fourier transform preserves separation of variables. Combining this observation with the result in Example 2.2.9, we conclude that the function $$f(x) = e^{-\pi \vert x\vert ^{2} }$$ defined on $$\mathbf{R}^{n}$$ is equal to its Fourier transform.

We now continue with some properties of the Fourier transform. Before we do this we introduce some notation. For a measurable function f on $$\mathbf{R}^{n}$$, $$x \in \mathbf{R}^{n}$$, and a > 0 we define the translation , dilation , and reflection of f by

(2.2.13)

Also recall the notation $$f_{a} = a^{-n}\delta ^{1/a}(f)$$ introduced in Definition 2.1.9.

Proposition 2.2.11.

Given f, $$g$$ in $$\mathcal{S}(\mathbf{R}^{n})$$ , $$y \in \mathbf{R}^{n}$$ , $$b \in \mathbf{C}$$ , $$\alpha$$ a multi-index, and t > 0, we have

(1)

$$\quad \big\|\widehat{f}\,\big\|_{L^{\infty }} \leq \big\| f\big\|_{L^{1}}$$ ,

(2)

$$\quad \widehat{f + g} =\widehat{ f} +\widehat{ g}$$ ,

(3)

$$\quad \widehat{bf} = b\widehat{f}$$ ,

(4)

$$\quad \widehat{\widetilde{f}\,\,} =\widetilde{\widehat{ f}\,\,}$$ ,

(5)

$$\quad \widehat{\overline{f\,}} = \overline{\widetilde{\widehat{f}\,\,}}$$ ,

(6)

$$\quad \widehat{\tau ^{y}f}(\xi ) = e^{-2\pi iy\cdot \xi }\widehat{f}(\xi )$$ ,

(7)

$$\quad (e^{2\pi ix\cdot y}f(x))\,\,\widehat{\,}\,\,(\xi ) =\tau ^{y}(\widehat{f}\,)(\xi )$$ ,

(8)

$$\quad (\delta ^{t}f)\,\,\widehat{\,}\, = t^{-n}\delta ^{t^{-1} }\widehat{f} = (\widehat{f}\,)_{t}\!$$ ,

(9)

$$\quad (\partial ^{\alpha }f)\,\,\widehat{\,}\,\,(\xi ) = (2\pi i\xi )^{\alpha }\widehat{f}(\xi )$$ ,

(10)

$$\quad (\partial ^{\alpha }\widehat{f}\,)(\xi ) = ((-2\pi ix)^{\alpha }f(x))\,\,\widehat{\,}\,\,(\xi )$$ ,

(11)

$$\quad \widehat{f} \in \mathcal{S}$$ ,

(12)

$$\quad \widehat{f {\ast} g} =\widehat{ f}\,\widehat{g}$$ ,

(13)

$$\quad \widehat{f \circ A}(\xi ) =\widehat{ f}(A\xi )$$ , where A is an orthogonal matrix and ξ is a column vector.

Proof.

Property (1) follows directly from Definition 2.2.8. Properties (2)–(5) are trivial. Properties (6)–(8) require a suitable change of variables but they are omitted. Property (9) is proved by integration by parts (which is justified by the rapid decay of the integrands):

To prove (10), let $$\alpha = e_{j} = (0,\ldots,1,\ldots,0)$$, where all entries are zero except for the jth entry, which is 1. Since

(2.2.14)

as h → 0 and the preceding function is bounded by C | x | for all h and ξ, the Lebesgue dominated convergence theorem implies that the integral of the function in (2.2.14) with respect to the measure f(x)dx converges to zero. This proves (10) for α = e j . For other α’s use induction. To prove (11) we use (9), (10), and (1) in the following way:

Identity (12) follows from the following calculation:

where the application of Fubini’s theorem is justified by the absolute convergence of the integrals. Finally, we prove (13). We have

where we used the change of variables y = Ax and the fact that | detA |  = 1. □ 

Corollary 2.2.12.

The Fourier transform of a radial function is radial. Products and convolutions of radial functions are radial.

Proof.

Let ξ ₁, ξ ₂ in $$\mathbf{R}^{n}$$ with $$\vert \xi _{1}\vert = \vert \xi _{2}\vert $$. Then for some orthogonal matrix A we have $$A\xi _{1} =\xi _{2}$$. Since f is radial, we have $$f = f \circ A$$. Then

where we used (13) in Proposition 2.2.11 to justify the second equality. Products and convolutions of radial functions are easily seen to be radial. □ 

2.2.3 The Inverse Fourier Transform and Fourier Inversion

We now define the inverse Fourier transform.

Definition 2.2.13.

Given a Schwartz function f, we define

for all $$x \in \mathbf{R}^{n}$$. The operation

is called the inverse Fourier transform.

It is straightforward that the inverse Fourier transform shares the same properties as the Fourier transform. One may want to list (and prove) properties for the inverse Fourier transform analogous to those in Proposition 2.2.11.

We now investigate the relation between the Fourier transform and the inverse Fourier transform. In the next theorem, we prove that one is the inverse operation of the other. This property is referred to as Fourier inversion.

Theorem 2.2.14.

Given f, g, and h in $$\mathcal{S}(\mathbf{R}^{n})$$ , we have

(1)

$$\int _{\mathbf{R}^{n}}f(x)\widehat{g}(x)\,dx =\int _{\mathbf{R}^{n}}\widehat{f}(x)g(x)\,dx\!$$ ,

(2)

(Fourier Inversion) $$\quad (\widehat{f}\,)\,^{\vee }\, = f = (f\,^{\vee }\,)^{\wedge }$$ ,

(3)

(Parseval’s relation) $$\int _{\mathbf{R}^{n}}f(x)\overline{h}(x)\,dx =\int _{\mathbf{R}^{n}}\widehat{f}(\xi )\overline{\widehat{h}(\xi )}\,d\xi \!$$ ,

(4)

(Plancherel’s identity) $$\quad \big\|f\big\|_{L^{2}} =\big\|\widehat{ f}\,\big\|_{L^{2}} =\big\| f\,^{\vee }\,\big\|_{L^{2}}\!$$ ,

(5)

$$\int _{\mathbf{R}^{n}}f(x)h(x)\,dx =\int _{\mathbf{R}^{n}}\widehat{f}(x)h\,^{\vee }\,(x)\,dx\!$$ .

Proof.

(1) follows immediately from the definition of the Fourier transform and Fubini’s theorem. To prove (2) we use (1) with

By Proposition 2.2.11 (7) and (8) and Example 2.2.9, we have that

which is an approximate identity. Now (1) gives

(2.2.15)

Now let $$\varepsilon \rightarrow 0$$ in (2.2.15). The left-hand side of (2.2.15) converges to f(t) uniformly on compact sets by Theorem 1.​2.​19.​ The right-hand side of (2.2.15) converges to $$(\widehat{f}\,)\,^{\vee }\,(t)$$ as $$\varepsilon \rightarrow 0$$ by the Lebesgue dominated convergence theorem. We conclude that $$(\widehat{f}\,)\,^{\vee }\,\! = f$$ on $$\mathbf{R}^{n}$$. Replacing f by $$\widetilde{f}$$ and using the result just proved, we conclude that $$(f\,^{\vee }\,)^{\wedge } = f$$.

Note that if $$g = \overline{\widehat{h}}$$, then Proposition 2.2.11 (5) and identity (2) imply that $$\widehat{g} = \overline{h}$$. Then (3) follows from (1) by expressing h in terms of g. Identity (4) is a trivial consequence of (3). (Sometimes the polarized identity (3) is also referred to as Plancherel’s identity.) Finally, (5) easily follows from (1) and (2) with $$\widehat{g} = h$$. □ 

Next we have the following simple corollary of Theorem 2.2.14.

Corollary 2.2.15.

The Fourier transform is a homeomorphism from $$\mathcal{S}(\mathbf{R}^{n})$$ onto itself.

Proof.

The continuity of the Fourier transform (and its inverse) follows from Exercise 2.2.2, while Fourier inversion yields that this map is bijective. □ 

2.2.4 The Fourier Transform on $$L^{1} + L^{2}$$

We have defined the Fourier transform on $$\mathcal{S}(\mathbf{R}^{n})$$. We now extend this definition to the space $$L^{1}(\mathbf{R}^{n}) + L^{2}(\mathbf{R}^{n})$$.

We begin by observing that the Fourier transform given in Definition 2.2.8,

makes sense as a convergent integral for functions $$f \in L^{1}(\mathbf{R}^{n}).$$ This allows us to extend the definition of the Fourier transform on L ¹. Moreover, this operator satisfies properties (1)–(8) as well as (12) and (13) in Proposition 2.2.11, with f, g integrable. We also define the inverse Fourier transform on L ¹ by setting $$f\,^{\vee }\,(x) =\widehat{ f}(-x)$$ for $$f \in L^{1}(\mathbf{R}^{n})$$ and we note that analogous properties hold for it. One problem in this generality is that when f is integrable, one may not necessarily have $$(\widehat{f}\,)\,^{\vee }\, = f$$ a.e. This inversion is possible when $$\widehat{f}$$ is also integrable; see Exercise 2.2.6.

The integral defining the Fourier transform does not converge absolutely for functions in $$L^{2}(\mathbf{R}^{n})$$; however, the Fourier transform has a natural definition in this space accompanied by an elegant theory. In view of the result in Exercise 2.2.8, the Fourier transform is an L ² isometry on $$L^{1} \cap L^{2}$$, which is a dense subspace of L ². By density, there is a unique bounded extension of the Fourier transform on L ². Let us denote this extension by $$\mathcal{F}$$. Then $$\mathcal{F}$$ is also an isometry on L ², i.e.,

for all $$f \in L^{2}(\mathbf{R}^{n})$$, and any sequence of functions $$f_{N} \in L^{1}(\mathbf{R}^{n}) \cap L^{2}(\mathbf{R}^{n})$$ converging to a given f in $$L^{2}(\mathbf{R}^{n})$$ satisfies

(2.2.16)

as $$N \rightarrow \infty $$. In particular, the sequence of functions $$f_{N}(x) = f(x)\chi _{\vert x\vert \leq N}$$ yields that

(2.2.17)

converges to $$\mathcal{F}(f)(\xi )$$ in L ² as $$N \rightarrow \infty $$. If f is both integrable and square integrable, the expressions in (2.2.17) also converge to $$\widehat{f}(\xi )$$ pointwise. Also, in view of Theorem 1.​1.​11 and (2.2.16), there is a subsequence of $$\widehat{f_{N}}$$ that converges to $$\mathcal{F}(f)$$ pointwise a.e. Consequently, for f in $$L^{1}(\mathbf{R}^{n}) \cap L^{2}(\mathbf{R}^{n})$$ the expressions $$\widehat{f}$$ and $$\mathcal{F}(f)$$ coincide pointwise a.e. For this reason we often adopt the notation $$\widehat{f}$$ to denote the Fourier transform of functions f in L ² as well.

In a similar fashion, we let $$\mathcal{F}^{{\prime}}$$ be the isometry on $$L^{2}(\mathbf{R}^{n})$$ that extends the operator $$f\mapsto f\,^{\vee }\,$$, which is an L ² isometry on $$L^{1} \cap L^{2}$$; the last statement follows by adapting the result of Exercise 2.2.8 to the inverse Fourier transform. Since $$\varphi \,^{\vee }\,(x) =\widehat{\varphi } (-x)$$ for $$\varphi$$ in the Schwartz class, which is dense in L ² (Exercise 2.2.5), it follows that $$\mathcal{F}^{{\prime}}(f)(x) = \mathcal{F}(f)(-x)$$ for all $$f \in L^{2}$$ and almost all $$x \in \mathbf{R}^{n}$$. The operators $$\mathcal{F}$$ and $$\mathcal{F}^{{\prime}}$$ are L ²-isometries that satisfy $$\mathcal{F}^{{\prime}}\circ \mathcal{F} = \mathcal{F}\circ \mathcal{F}^{{\prime}} = \text{Id}$$ on the Schwartz space. By density this identity also holds for L ² functions and implies that $$\mathcal{F}$$ and $$\mathcal{F}^{{\prime}}$$ are injective and surjective mappings from L ² to itself; consequently, $$\mathcal{F}^{{\prime}}$$ coincides with the inverse operator $$\mathcal{F}^{-1}$$ of $$\mathcal{F}:\, L^{2} \rightarrow L^{2}$$, and Fourier inversion

holds on L ².

Having set down the basic facts concerning the action of the Fourier transform on L ¹ and L ², we extend its definition on L p for 1 < p < 2. Given a function f in $$L^{p}(\mathbf{R}^{n})$$, with 1 < p < 2, we define $$\widehat{f} =\widehat{ f_{1}} +\widehat{ f_{2}}$$, where $$f_{1} \in L^{1}(\mathbf{R}^{n})$$, $$f_{2} \in L^{2}(\mathbf{R}^{n})$$, and $$f = f_{1} + f_{2}$$; we may take, for instance, $$f_{1} = f\chi _{\vert f\vert >1}$$ and $$f_{2} = f\chi _{\vert f\vert \leq 1}$$. The definition of $$\widehat{f}$$ is independent of the choice of f ₁ and f ₂, for if $$f_{1} + f_{2} = h_{1} + h_{2}$$ for $$f_{1},h_{1} \in L^{1}(\mathbf{R}^{n})$$ and $$f_{2},h_{2} \in L^{2}(\mathbf{R}^{n})$$, we have $$f_{1} - h_{1} = h_{2} - f_{2} \in L^{1}(\mathbf{R}^{n}) \cap L^{2}(\mathbf{R}^{n})$$. Since these functions are equal on $$L^{1}(\mathbf{R}^{n}) \cap L^{2}(\mathbf{R}^{n})$$, their Fourier transforms are also equal, and we obtain $$\widehat{f_{1}} -\widehat{ h_{1}} =\widehat{ h_{2}} -\widehat{ f_{2}}$$, which yields $$\widehat{f_{1} + f_{2}} =\widehat{ h_{1} + h_{2}}$$. We have the following result concerning the action of the Fourier transform on $$L^{p}$$.

Proposition 2.2.16.

(Hausdorff–Young inequality) For every function f in $$L^{p}(\mathbf{R}^{n})$$ we have the estimate

whenever $$1 \leq p \leq 2$$ .

Proof.

This follows easily from Theorem 1.​3.​4.​ Interpolate between the estimates $$\|\widehat{f}\,\|_{L^{\infty }} \leq \| f\|_{L^{1}}$$ (Proposition 2.2.11 (1)) and $$\|\widehat{f}\,\|_{L^{2}} \leq \| f\|_{L^{2}}$$ to obtain $$\|\widehat{f}\,\|_{L^{p'}} \leq \| f\|_{L^{p}}$$. We conclude that the Fourier transform is a bounded operator from $$L^{p}(\mathbf{R}^{n})$$ to $$L^{p'}(\mathbf{R}^{n})$$ with norm at most 1 when $$1 \leq p \leq 2$$. □ 

Next, we are concerned with the behavior of the Fourier transform at infinity.

Proposition 2.2.17.

(Riemann–Lebesgue lemma) For a function f in $$L^{1}(\mathbf{R}^{n})$$ we have that

Proof.

Consider the function $$\chi _{[a,b]}$$ on $$\mathbf{R}$$. A simple computation gives

which tends to zero as $$\vert \xi \vert \rightarrow \infty $$. Likewise, if $$g =\prod _{ j=1}^{n}\chi _{[a_{j},b_{j}]}$$ on $$\mathbf{R}^{n}$$, then

Given a $$\xi = (\xi _{1},\ldots,\xi _{n})\neq 0$$, there is j ₀ such that $$\vert \xi _{j_{0}}\vert > \vert \xi \vert /\sqrt{n}$$. Then

which also tends to zero as $$\vert \xi \vert \rightarrow \infty $$ in $$\mathbf{R}^{n}$$.

Given a general integrable function f on $$\mathbf{R}^{n}$$ and $$\varepsilon > 0$$, there is a simple function h, which is a finite linear combination of characteristic functions of rectangles (like g), such that $$\|f - h\|_{L^{1}} < \frac{\varepsilon } {2}$$. Then there is an M is such that for | ξ |  > M we have $$\vert \widehat{h}(\xi )\vert < \frac{\varepsilon } {2}$$. It follows that

provided | ξ |  > M. This implies that $$\vert \widehat{f}(\xi )\vert \rightarrow 0$$ as $$\vert \xi \vert \rightarrow \infty $$.

A different proof can be given by taking the function h in the preceding paragraph to be a Schwartz function and using that Schwartz functions are dense in $$L^{1}(\mathbf{R}^{n})$$; see Exercise 2.2.5 about the last assertion. □ 

We end this section with an example that illustrates some of the practical uses of the Fourier transform.

Example 2.2.18.

We would like to find a Schwartz function $$f(x_{1},x_{2},x_{3})$$ on $$\mathbf{R}^{3}$$ that satisfies the partial differential equation

Taking the Fourier transform on both sides of this identity and using Proposition 2.2.11 (2), (9) and the result of Example 2.2.9, we obtain

Let $$p(\xi ) = p(\xi _{1},\xi _{2},\xi _{3})$$ be the polynomial inside the square brackets. We observe that $$p(\xi )$$ has no real zeros and we may therefore write

In general, let

be a polynomial in $$\mathbf{R}^{n}$$ with constant complex coefficients $$C_{\alpha }$$ indexed by multi-indices $$\alpha$$. If $$P(2\pi i\xi )$$ has no real zeros, and u is in $$\mathcal{S}(\mathbf{R}^{n})$$, then the partial differential equation

is solved as before to give

Since $$P(2\pi i\xi )$$ has no real zeros and $$u \in \mathcal{S}(\mathbf{R}^{n})$$, the function

is smooth and therefore a Schwartz function. Then f is also in $$\mathcal{S}(\mathbf{R}^{n})$$ by Proposition 2.2.11 (11).

Exercises

2.2.1. (a) Construct a Schwartz function supported in the unit ball of $$\mathbf{R}^{n}$$.

(b) Construct a $$\mathcal{C}_{0}^{\infty }(\mathbf{R}^{n})$$ function equal to 1 on the annulus $$1 \leq \vert x\vert \leq 2$$ and vanishing off the annulus $$1/2 \leq \vert x\vert \leq 4$$.

(c) Construct a nonnegative nonzero Schwartz function $$f$$ on $$\mathbf{R}^{n}$$ whose Fourier transform is nonnegative and compactly supported.

[Hint: Part (a): Try the construction in dimension one first using the $$\mathcal{C}^{\infty }$$ function $$\eta (x) = e^{-1/x}$$ for x > 0 and η(x) = 0 for x < 0. Part (c): Take $$f = \vert \phi {\ast}\widetilde{\phi }\vert ^{2}$$, where $$\widehat{\phi }$$ is odd, real-valued, and compactly supported; here $$\widetilde{\phi }(x) =\phi (-x)$$.]

2.2.2.

If $$f_{k},f \in \mathcal{S}(\mathbf{R}^{n})$$ and $$f_{k} \rightarrow f$$ in $$\mathcal{S}(\mathbf{R}^{n})$$, then $$\widehat{f_{k}} \rightarrow \widehat{ f}$$ and $$f_{k}\,^{\vee }\,\rightarrow f\,^{\vee }\,$$ in $$\mathcal{S}(\mathbf{R}^{n})$$.

2.2.3.

Find the spectrum (i.e., the set of all eigenvalues of the Fourier transform ), that is, all complex numbers $$\lambda$$ for which there exist nonzero functions f such that

[Hint: Apply the Fourier transform three times to the preceding identity. Consider the functions $$xe^{-\pi x^{2} }$$, $$(a + bx^{2})e^{-\pi x^{2} }$$, and $$(cx + dx^{3})e^{-\pi x^{2} }$$ for suitable a, b, c, d to show that all fourth roots of unity are indeed eigenvalues of the Fourier transform.]

2.2.4. Use the idea of the proof of Proposition 2.2.7 to show that if the functions $$f$$, $$g$$ defined on $$\mathbf{R}^{n}$$ satisfy $$\vert f(x)\vert \leq A(1 + \vert x\vert )^{-M}$$ and | g(x) | ≤ B(1 + | x | )−N for some M, N > n, then

where L = min(N, M) and C = C(N, M) > 0.

2.2.5.

Show that $$\mathcal{C}_{0}^{\infty }(\mathbf{R}^{n})$$ is dense on $$L^{p}(\mathbf{R}^{n})$$ for $$0 < p < \infty $$ but not for $$p = \infty $$.

[Hint: Use a smooth approximate identity when p ≥ 1. Reduce the case p < 1 to p = 1.]

2.2.6.

(a) Prove that if $$f \in L^{1}$$, then $$\widehat{f}$$ is uniformly continuous on $$\mathbf{R}^{n}$$.

(b) Prove that for $$f,g \in L^{1}(\mathbf{R}^{n})$$ we have

(c) Take $$\widehat{g}(x) =\varepsilon ^{-n}e^{-\pi \varepsilon ^{-2}\vert x-t\vert ^{2} }$$ in (b) and let $$\varepsilon \rightarrow 0$$ to prove that if f and $$\widehat{f}$$ are both in L ¹, then $$(\widehat{f}\,)\,^{\vee }\, = f$$ a.e. This fact is called Fourier inversion on L ¹.

2.2.7.

(a) Prove that if a function f in $$L^{1}(\mathbf{R}^{n}) \cap L^{\infty }(\mathbf{R}^{n})$$ is continuous at 0, then

(b) Let $$f \in L^{1}(\mathbf{R}^{n}) \cap L^{\infty }(\mathbf{R}^{n})$$ be continuous at zero and satisfy $$\widehat{f} \geq 0$$. Show that $$\widehat{f}$$ is in L ¹ and conclude that Fourier inversion holds at zero $$f(0) =\|\widehat{ f}\,\|_{L^{1}}$$, and also $$f = (\widehat{f}\,)\,^{\vee }\,$$ a.e. in general.

[Hint: Part (a): Let $$g(x) = e^{-\pi \vert \varepsilon x\vert ^{2} }$$ in Exercise 2.2.6(b) and use Theorem 1.​2.​19 (2).]

2.2.8.

Given f in $$L^{1}(\mathbf{R}^{n}) \cap L^{2}(\mathbf{R}^{n})$$, without appealing to density, prove that

[Hint: Let $$h = f {\ast}\overline{\widetilde{f}\,}$$, where $$\widetilde{f}(x) = f(-x)$$ and the bar indicates complex conjugation. Then $$h \in L^{1}(\mathbf{R}^{n})\cap L^{\infty }(\mathbf{R}^{n})$$, $$\widehat{h} = \vert \widehat{f}\vert ^{2} \geq 0$$, and h is continuous at zero. Exercise 2.2.7(b) yields $$\|\widehat{f}\,\|_{L^{2}}^{2} =\|\widehat{ h}\|_{L^{1}} = h(0) =\int _{\mathbf{R}^{n}}f(x)\overline{\widetilde{f}(-x)}\,dx =\| f\|_{L^{2}}^{2}$$.]

2.2.9.

(a) Prove that for all $$0 <\varepsilon < t < \infty $$ we have

(b) If f is an odd L ¹ function on the line, conclude that for all $$t >\varepsilon > 0$$ we have

(c) Let g(ξ) be a continuous odd function that is equal to 1∕log(ξ) for ξ ≥ 2. Show that there does not exist an L ¹ function whose Fourier transform is g.

2.2.10.

Let f be in $$L^{1}(\mathbf{R})$$. Prove that

[Hint: For $$x \in (-\infty,0)$$ use the change of variables $$u = x -\frac{1} {x}$$ or $$x = \frac{1} {2}\big(u -\sqrt{4 + u^{2}}\,\big)$$. For x ∈ (0, ∞) use the change of variables $$u = x -\frac{1} {x}$$ or $$x = \frac{1} {2}\big(u + \sqrt{4 + u^{2}}\,\big)$$.]

2.2.11.

(a) Use Exercise 2.2.10 with $$f(x) = e^{-tx^{2} }$$ to obtain the subordination identity

(b) Set t = π | x | and integrate with respect to $$e^{-2\pi i\xi \cdot x}dx$$ to prove that

This calculation gives the Fourier transform of the Poisson kernel.

2.2.12. Let $$1 \leq p \leq \infty $$ and let p′ be its dual index. (a) Prove that Schwartz functions f on the line satisfy the estimate

(b) Prove that all Schwartz functions f on $$\mathbf{R}^{n}$$ satisfy the estimate

where the sum is taken over all pairs of multi-indices $$\alpha$$ and $$\beta$$ whose sum has size n.

[Hint: Part (a): Write $$f(x)^{2} =\int _{ -\infty }^{x} \tfrac{d} {dt}f(t)^{2}\,dt$$.]

2.2.13. The uncertainty principle says that the position and the momentum of a particle cannot be simultaneously localized. Prove the following inequality, which presents a quantitative version of this principle:

where f is a Schwartz function on $$\mathbf{R}^{n}$$ (or an L ² function with sufficient decay at infinity).

[Hint: Let y be in $$\mathbf{R}^{n}$$. Start with

integrate by parts, apply the Cauchy–Schwarz inequality, Plancherel’s identity, and the identity $$\sum _{j=1}^{n}\vert \widehat{\partial _{j}f}(\xi )\vert ^{2} = 4\pi ^{2}\vert \xi \vert ^{2}\vert \widehat{f}(\xi )\vert ^{2}$$ for all $$\xi \in \mathbf{R}^{n}$$. Then replace f(x) by $$f(x)e^{2\pi ix\cdot z}$$.]

2.2.14. Let $$-\infty <\alpha < \frac{n} {2} <\beta < +\infty $$. Prove the validity of the following inequality:

for some constant $$C = C(n,\alpha,\beta )$$ independent of g.

[Hint: First prove $$\|g\|_{L^{1}} \leq C\|\,\vert x\vert ^{\alpha }g(x)\|_{L^{2}} +\|\, \vert x\vert ^{\beta }g(x)\|_{L^{2}}$$ and then replace g(x) by $$g(\lambda x)$$ for some suitable $$\lambda > 0$$.]

2.3 The Class of Tempered Distributions

The fundamental idea of the theory of distributions is that it is generally easier to work with linear functionals acting on spaces of “nice” functions than to work with “bad” functions directly. The set of “nice” functions we consider is closed under the basic operations in analysis, and these operations are extended to distributions by duality. This wonderful interpretation has proved to be an indispensable tool that has clarified many situations in analysis.

2.3.1 Spaces of Test Functions

We recall the space $$\mathcal{C}_{0}^{\infty }(\mathbf{R}^{n})$$ of all smooth functions with compact support, and $$\mathcal{C}^{\infty }(\mathbf{R}^{n})$$ of all smooth functions on $$\mathbf{R}^{n}$$. We are mainly interested in the three spaces of “nice” functions on $$\mathbf{R}^{n}$$ that are nested as follows:

Here $$\mathcal{S}(\mathbf{R}^{n})$$ is the space of Schwartz functions introduced in Section 2.2.

Definition 2.3.1.

We define convergence of sequences in these spaces. We say that

It follows that convergence in $$\mathcal{C}_{0}^{\infty }(\mathbf{R}^{n})$$ implies convergence in $$\mathcal{S}(\mathbf{R}^{n})$$, which in turn implies convergence in $$\mathcal{C}^{\infty }(\mathbf{R}^{n})$$.

Example 2.3.2.

Let $$\varphi$$ be a nonzero $$\mathcal{C}_{0}^{\infty }$$ function on $$\mathbf{R}$$. We call such functions smooth bumps. Define the sequence of smooth bumps $$\varphi _{k}(x) =\varphi (x - k)/k$$. Then $$\varphi _{k}(x)$$ does not converge to zero in $$\mathcal{C}_{0}^{\infty }(\mathbf{R})$$, even though $$\varphi _{k}$$ (and all of its derivatives) converge to zero uniformly. Furthermore, we see that $$\varphi _{k}$$ does not converge to any function in $$\mathcal{S}(\mathbf{R})$$. Clearly $$\varphi _{k} \rightarrow 0$$ in $$\mathcal{C}^{\infty }(\mathbf{R}).$$

The space $$\mathcal{C}^{\infty }(\mathbf{R}^{n})$$ is equipped with the family of seminorms

(2.3.1)

where $$\alpha$$ ranges over all multi-indices and $$N$$ ranges over $$\mathbf{Z}^{+}$$. It can be shown that $$\mathcal{C}^{\infty }(\mathbf{R}^{n})$$ is complete with respect to this countable family of seminorms, i.e., it is a Fréchet space. However, it is true that $$\mathcal{C}_{0}^{\infty }(\mathbf{R}^{n})$$ is not complete with respect to the topology generated by this family of seminorms.

The topology of $$\mathcal{C}_{0}^{\infty }$$ given in Definition 2.3.1 is the inductive limit topology, and under this topology it is complete. Indeed, letting $$\mathcal{C}_{0}^{\infty }(B(0,k))$$ be the space of all smooth functions with support in B(0, k), then $$\mathcal{C}_{0}^{\infty }(\mathbf{R}^{n})$$ is equal to $$\bigcup _{k=1}^{\infty }\mathcal{C}_{0}^{\infty }(B(0,k))$$ and each space $$\mathcal{C}_{0}^{\infty }(B(0,k))$$ is complete with respect to the topology generated by the family of seminorms $$\widetilde{\rho }_{\alpha,N};$$ hence so is $$\mathcal{C}_{0}^{\infty }(\mathbf{R}^{n})$$. Nevertheless, $$\mathcal{C}_{0}^{\infty }(\mathbf{R}^{n})$$ is not metrizable. Details on the topologies of these spaces can be found in [286].

2.3.2 Spaces of Functionals on Test Functions

The dual spaces (i.e., the spaces of continuous linear functionals on the sets of test functions) we introduced is denoted by

By definition of the topologies on the dual spaces, we have

The dual spaces are nested as follows:

Definition 2.3.3.

Elements of the space $$\mathcal{D}'(\mathbf{R}^{n})$$ are called distributions. Elements of $$\mathcal{S}'(\mathbf{R}^{n})$$ are called tempered distributions. Elements of the space $$\mathcal{E}'(\mathbf{R}^{n})$$ are called distributions with compact support.

Before we discuss some examples, we give alternative characterizations of distributions, which are very useful from the practical point of view. The action of a distribution u on a test function f is represented in either one of the following two ways:

Proposition 2.3.4.

(a) A linear functional u on $$\mathcal{C}_{0}^{\infty }(\mathbf{R}^{n})$$ is a distribution if and only if for every compact $$K \subseteq \mathbf{R}^{n}$$ , there exist C > 0 and an integer m such that

(2.3.2)

(b) A linear functional u on $$\mathcal{S}(\mathbf{R}^{n})$$ is a tempered distribution if and only if there exist C > 0 and k, m integers such that

(2.3.3)

(c) A linear functional $$u$$ on $$\mathcal{C}^{\infty }(\mathbf{R}^{n})$$ is a distribution with compact support if and only if there exist C > 0 and N, m integers such that

(2.3.4)

The seminorms $$\rho _{\alpha,\beta }$$ and $$\widetilde{\rho }_{\alpha,N}$$ are defined in(2.2.6) and (2.3.1), respectively.

Proof.

We prove only (2.3.3), since the proofs of (2.3.2) and (2.3.4) are similar. It is clear that (2.3.3) implies continuity of u. Conversely, it was pointed out in Section 2.2 that the family of sets $$\{f \in \mathcal{S}(\mathbf{R}^{n}):\,\,\rho _{\alpha,\beta }(f) <\delta \},$$ where $$\alpha$$, $$\beta$$ are multi-indices and $$\delta > 0$$, forms a subbasis for the topology of $$\mathcal{S}$$. Thus if u is a continuous functional on $$\mathcal{S}$$, there exist integers k, m and a $$\delta > 0$$ such that

(2.3.5)

We see that (2.3.3) follows from (2.3.5) with $$C = 1/\delta$$. □ 

Example 2.3.5.

We now discuss some important examples.

1.

The Dirac mass at the origin $$\delta _{0}$$. This is defined for $$\varphi \in \mathcal{C}^{\infty }(\mathbf{R}^{n})$$ by

We claim that $$\delta _{0}$$ is in $$\mathcal{E}'$$. To see this we observe that if $$\varphi _{k} \rightarrow \varphi$$ in $$\mathcal{C}^{\infty }$$ then $$\big\langle \delta _{0},\varphi _{k}\big\rangle \rightarrow \big\langle \delta _{0},\varphi \big\rangle$$. The Dirac mass at a point $$a \in \mathbf{R}^{n}$$ is defined similarly by

2.

Some functions g can be thought of as distributions via the identification $$g\mapsto L_{g}$$, where L g is the functional

Here are some examples: The function 1 is in $$\mathcal{S}'$$ but not in $$\mathcal{E}'$$. Compactly supported integrable functions are in $$\mathcal{E}'$$. The function $$e^{\vert x\vert ^{2} }$$ is in $$\mathcal{D}'$$ but not in $$\mathcal{S}'$$.

3.

Functions in $$L_{\mbox{ loc}}^{1}$$ are distributions. To see this, first observe that if $$g \in L_{\mbox{ loc}}^{1},$$ then the integral

is well defined for all $$\varphi \in \mathcal{D}$$ and satisfies $$\vert L_{g}(\varphi )\vert \leq \big (\int _{K}\vert g(x)\vert \,dx\big)\|\varphi \|_{L^{\infty }}$$ for all smooth functions $$\varphi$$ supported in the compact set K.

4.

Functions in $$L^{p}$$, $$1 \leq p \leq \infty $$, are tempered distributions, but may not in $$\mathcal{E}'$$ unless they have compact support.

5.

Any finite Borel measure μ is a tempered distribution via the identification

To see this, observe that $$\varphi _{k} \rightarrow \varphi$$ in $$\mathcal{S}$$ implies that $$L_{\mu }(\varphi _{k}) \rightarrow L_{\mu }(\varphi )$$. Finite Borel measures may not be distributions with compact support.

6.

Every function g that satisfies $$\vert g(x)\vert \leq C(1 + \vert x\vert )^{k},$$ for some real number k, is a tempered distribution. To see this, observe that

where m > n + k and the expression $$\sup _{x\in \mathbf{R}^{n}}(1 + \vert x\vert )^{m}\vert \varphi (x)\vert $$ is bounded by a finite sum of Schwartz seminorms $$\rho _{\alpha,\beta }(\varphi )$$.

7.

The function log | x | is a tempered distribution; indeed for any $$\varphi \in \mathcal{S}(\mathbf{R}^{n})$$, the integral of $$\varphi (x)\log \vert x\vert $$ is bounded by a finite number of Schwartz seminorms of $$\varphi$$. More generally, any function that is integrable on a ball | x | ≤ M and for some C > 0 satisfies | g(x) | ≤ C(1 + | x | ) k for | x | ≥ M, is a tempered distribution.

8.

Here is an example of a compactly supported distribution on $$\mathbf{R}$$ that is neither a locally integrable function nor a finite Borel measure:

We have that $$\vert \langle u,\varphi \rangle \vert \leq 2\|\varphi '\|_{L^{\infty }([-1,1])}$$ and notice that $$\|\varphi '\|_{L^{\infty }([-1,1])}$$ is a $$\widetilde{\rho }_{\alpha,N}$$ seminorm of φ.

2.3.3 The Space of Tempered Distributions

Having set down the basic definitions of distributions, we now focus our study on the space of tempered distributions. These distributions are the most useful in harmonic analysis. The main reason for this is that the subject is concerned with boundedness of translation-invariant operators, and every such bounded operator from $$L^{p}(\mathbf{R}^{n})$$ to $$L^{q}(\mathbf{R}^{n})$$ is given by convolution with a tempered distribution. This fact is shown in Section 2.5.

Suppose that f and g are Schwartz functions and $$\alpha$$ a multi-index. Integrating by parts $$\vert \alpha \vert $$ times, we obtain

(2.3.6)

If we wanted to define the derivative of a tempered distribution u, we would have to give a definition that extends the definition of the derivative of the function and that satisfies (2.3.6) for g in $$\mathcal{S}'$$ and $$f \in \mathcal{S}$$ if the integrals in (2.3.6) are interpreted as actions of distributions on functions. We simply use equation (2.3.6) to define the derivative of a distribution.

Definition 2.3.6.

Let $$u \in \mathcal{S}'$$ and $$\alpha$$ a multi-index. Define

(2.3.7)

If u is a function, the derivatives of u in the sense of distributions are called distributional derivatives.

In view of Theorem 2.2.14, it is natural to give the following:

Definition 2.3.7.

Let $$u \in \mathcal{S}'$$. We define the Fourier transform $$\widehat{u}$$ and the inverse Fourier transform $$u\,^{\vee }\,$$ of a tempered distribution $$u$$ by

(2.3.8)

for all $$f$$ in $$\mathcal{S}$$.

Example 2.3.8.

We observe that $$\widehat{\delta _{0}} = 1$$. More generally, for any multi-index $$\alpha$$ we have

To see this, observe that for all $$f \in \mathcal{S}$$ we have

This calculation indicates that $$(\partial ^{\alpha }\delta _{0})^{\wedge }\,$$ can be identified with the function $$(2\pi ix)^{\alpha }$$.

Example 2.3.9.

Recall that for $$x_{0} \in \mathbf{R}^{n}$$, $$\delta _{x_{0}}(f) =\big\langle \delta _{x_{0}},f\big\rangle = f(x_{0})$$. Then

that is, $$\widehat{\delta _{x_{0}}}$$ can be identified with the function $$x\mapsto e^{-2\pi ix\cdot x_{0}}$$. In particular, $$\widehat{\delta _{0}} = 1$$.

Example 2.3.10.

The function $$e^{\vert x\vert ^{2} }$$ is not in $$\mathcal{S}'(\mathbf{R}^{n})$$ and therefore its Fourier transform is not defined as a distribution. However, the Fourier transform of any locally integrable function with polynomial growth at infinity is defined as a tempered distribution.

Now observe that the following are true whenever f, g are in $$\mathcal{S}$$.

(2.3.9)

for all $$t \in \mathbf{R}^{n}$$ and a > 0. Recall now the definitions of τ t , $$\delta ^{a}$$, and $$\,\,\,\widetilde{}\,\,$$ given in (2.2.13). Motivated by (2.3.9), we give the following:

Definition 2.3.11.

The translation τ t u, the dilation $$\delta ^{a}u$$, and the reflection $$\widetilde{u}$$ of a tempered distribution u are defined as follows:

(2.3.10)

(2.3.11)

(2.3.12)

for all $$t \in \mathbf{R}^{n}$$ and a > 0. Let A be an invertible matrix. The composition of a distribution u with an invertible matrix A is the distribution

(2.3.13)

where $$\varphi ^{A^{-1} }(x) =\varphi (A^{-1}x)$$.

It is easy to see that the operations of translation, dilation, reflection, and differentiation are continuous on tempered distributions.

Example 2.3.12.

The Dirac mass at the origin $$\delta _{0}$$ is equal to its reflection, while $$\delta ^{a}\delta _{0} = a^{-n}\delta _{0}$$. Also, $$\tau ^{x}\delta _{0} =\delta _{x}$$ for any $$x \in \mathbf{R}^{n}$$.

Now observe that for f, g, and h in $$\mathcal{S}$$ we have

(2.3.14)

Motivated by (2.3.14), we define the convolution of a function with a tempered distribution as follows:

Definition 2.3.13.

Let $$u \in \mathcal{S}'$$ and $$h \in \mathcal{S}$$. Define the convolution h ∗ u by

(2.3.15)

Example 2.3.14.

Let $$u =\delta _{x_{0}}$$ and $$f \in \mathcal{S}$$. Then $$f {\ast}\delta _{x_{0}}$$ is the function $$x\mapsto f(x - x_{0})$$, for when $$h \in \mathcal{S}$$, we have

It follows that convolution with $$\delta _{0}$$ is the identity operator.

We now define the product of a function and a distribution.

Definition 2.3.15.

Let $$u \in \mathcal{S}'$$ and let h be a $$\mathcal{C}^{\infty }$$ function that has at most polynomial growth at infinity and the same is true for all of its derivatives. This means that for all $$\alpha$$ it satisfies $$\vert (\partial ^{\alpha }h)(x)\vert \leq C_{\alpha }(1 + \vert x\vert )^{k_{\alpha }}$$ for some $$C_{\alpha },k_{\alpha } > 0$$. Then define the product hu of h and u by

(2.3.16)

Note that hf is in $$\mathcal{S}$$ and thus (2.3.16) is well defined. The product of an arbitrary $$\mathcal{C}^{\infty }$$ function with a tempered distribution is not defined.

We observe that if a function g is supported in a set K, then for all $$f \in \mathcal{C}_{0}^{\infty }(K^{c})$$ we have

(2.3.17)

Moreover, the support of g is the intersection of all closed sets K with the property (2.3.17) for all f in $$\mathcal{C}_{0}^{\infty }(K^{c})$$. Motivated by the preceding observation we give the following:

Definition 2.3.16.

Let u be in $$\mathcal{D}'(\mathbf{R}^{n})$$. The support of u (supp ​u) is the intersection of all closed sets K with the property

(2.3.18)

Distributions with compact support are exactly those whose support (as defined in the previous definition) is a compact set. To prove this assertion, we start with a distribution u with compact support as defined in Definition 2.3.3. Then there exist C, N, m > 0 such that (2.3.4) holds. For a $$\mathcal{C}^{\infty }$$ function f whose support is contained in B(0, N) c , the expression on the right in (2.3.4) vanishes and we must therefore have $$\langle u,f\rangle = 0$$. This shows that the support of u is contained in $$\overline{B(0,N)}$$ hence it is bounded, and since it is already closed (as an intersection of closed sets), it must be compact. Conversely, if the support of u as defined in Definition 2.3.16 is a compact set, then there exists an N > 0 such that supp ​u is contained in B(0, N). We take a smooth function η that is equal to 1 on B(0, N) and vanishes off B(0, N + 1). Then for $$h \in \mathcal{C}_{0}^{\infty }$$ the support of h(1 −η) does not meet the support of u, and we must have

The distribution u can be thought of as an element of $$\mathcal{E}'$$ by defining for $$f \in \mathcal{C}^{\infty }(\mathbf{R}^{n})$$

Taking m to be the integer that corresponds to the compact set $$K = \overline{B(0,N + 1)}$$ in (2.3.2), and using that the L ∞ norm of $$\partial ^{\alpha }(f\eta )$$ is controlled by a finite sum of seminorms $$\widetilde{\rho }_{\alpha,N+1}(f)$$ with | α | ≤ m, we obtain the validity of (2.3.4) for $$f \in \mathcal{C}^{\infty }$$.

Example 2.3.17.

The support of the Dirac mass at x ₀ is the set {x ₀}.

Along the same lines, we give the following definition:

Definition 2.3.18.

We say that a distribution u in $$\mathcal{D}'(\mathbf{R}^{n})$$ coincides with the function h on an open set $$\varOmega$$ if

(2.3.19)

When (2.3.19) occurs we often say that u agrees with h away from $$\varOmega ^{c}$$.

This definition implies that the support of the distribution u − h is contained in the set $$\varOmega ^{c}$$.

Example 2.3.19.

The distribution $$\vert x\vert ^{2} +\delta _{a_{1}} +\delta _{a_{2}}$$, where a ₁, a ₂ are in $$\mathbf{R}^{n}$$, coincides with the function | x | ² on any open set not containing the points a ₁ and a ₂. Also, the distribution in Example 2.3.5 (8) coincides with the function $$x^{-1}\chi _{\vert x\vert \leq 1}$$ away from the origin in the real line.

Having ended the streak of definitions regarding operations with distributions, we now discuss properties of convolutions and Fourier transforms.

Theorem 2.3.20.

If $$u \in \mathcal{S}'$$ and $$\varphi \in \mathcal{S}$$ , then $$\varphi {\ast}u$$ is a $$\mathcal{C}^{\infty }$$ function and

for all $$x \in \mathbf{R}^{n}$$ . Moreover, for all multi-indices $$\alpha$$ there exist constants $$C_{\alpha },k_{\alpha } > 0$$ such that

Furthermore, if u has compact support, then $$\varphi {\ast}u$$ is a Schwartz function.

Proof.

Let ψ be in $$\mathcal{S}(\mathbf{R}^{n})$$. We have

(2.3.20)

where the last step is justified by the continuity of u and by the fact that the Riemann sums of the inner integral in (2.3.20) converge to that integral in the topology of $$\mathcal{S}$$, a fact that will be justified later. This calculation identifies the function $$\varphi {\ast}u$$ as

(2.3.21)

We now show that $$(\varphi {\ast}u)(x)$$ is a $$\mathcal{C}^{\infty }$$ function. Let $$e_{j} = (0,\ldots,1,\ldots,0)$$ with 1 in the jth entry and zero elsewhere. Then

by the continuity of u and the fact that $$\big(\tau ^{-he_{j}}(\tau ^{x}\widetilde{\varphi }) -\tau ^{x}\widetilde{\varphi }\big)/h$$ tends to $$\partial _{j}\tau ^{x}\widetilde{\varphi } =\tau ^{x}(\partial _{j}\widetilde{\varphi })$$ in $$\mathcal{S}$$ as h → 0; see Exercise 2.3.5 (a). The same calculation for higher-order derivatives shows that $$\varphi {\ast}u \in \mathcal{C}^{\infty }$$ and that $$\partial ^{\gamma }(\varphi {\ast}u) = (\partial ^{\gamma }\varphi ) {\ast} u$$ for all multi-indices $$\gamma$$. It follows from (2.3.3) that for some C, m, and k we have

(2.3.22)

and this clearly implies that $$\partial ^{\alpha }(\varphi {\ast}u)$$ grows at most polynomially at infinity.

We now indicate why $$\varphi {\ast}u$$ is Schwartz whenever u has compact support. Applying estimate (2.3.4) to the function $$y\mapsto \varphi (x - y)$$ yields that

for some constants C, m, N. Since for | x | ≥ 2N we have

it follows that $$\varphi {\ast}u$$ decays rapidly at infinity. Since $$\partial ^{\gamma }(\varphi {\ast}u) = (\partial ^{\gamma }\varphi ) {\ast} u$$, the same argument yields that all the derivatives of $$\varphi {\ast}u$$ decay rapidly at infinity; hence $$\varphi {\ast}u$$ is a Schwartz function. Incidentally, this argument actually shows that any Schwartz seminorm of $$\varphi {\ast}u$$ is controlled by a finite sum of Schwartz seminorms of $$\varphi$$.

We now return to the point left open concerning the convergence of the Riemann sums in (2.3.20) in the topology of $$\mathcal{S}(\mathbf{R}^{n})$$. For each $$N = 1,2,\ldots$$, consider a partition of [−N, N] n into $$(2N^{2})^{n}$$ cubes Q m of side length 1∕N and let y m be the center of each Q m . For multi-indices $$\alpha,\beta$$, we must show that

converges to zero in $$L^{\infty }(\mathbf{R}^{n})$$ as $$N \rightarrow \infty $$. We have

for some $$\xi = y +\theta (y_{m} - y)$$, where θ ∈ [0, 1]. Distributing the gradient to both factors, we see that the last integrand is at most

for M large (pick $$M > 2\vert \alpha \vert $$), which in turn is at most

since $$\vert y\vert \leq \vert \xi \vert +\theta \vert y - y_{m}\vert \leq \vert \xi \vert + \sqrt{n}/N \leq \vert \xi \vert + 1$$ for $$N \geq \sqrt{n}$$. Inserting the estimate obtained for the integrand in the last displayed integral, we obtain

But the second integral in the preceding expression is bounded by

Using these estimates it is now easy to see that $$\lim _{N\rightarrow \infty }\sup _{x\in \mathbf{R}^{n}}\vert D_{N}(x)\vert = 0$$. □ 

Next we have the following important result regarding distributions with compact support:

Theorem 2.3.21.

If u is in $$\mathcal{E}'(\mathbf{R}^{n})$$ , then $$\widehat{u}$$ is a real analytic function on $$\mathbf{R}^{n}$$ . In particular, $$\widehat{u}$$ is a $$\mathcal{C}^{\infty }$$ function. Furthermore, $$\widehat{u}$$ and all of its derivatives have polynomial growth at infinity. Moreover, $$\widehat{u}$$ has a holomorphic extension on $$\mathbf{C}^{n}$$ .

Proof.

Given a distribution u with compact support and a polynomial p(ξ), the action of u on the $$\mathcal{C}^{\infty }$$ function $$\xi \mapsto p(\xi )e^{-2\pi ix\cdot \xi }$$ is a well defined function of x, which we denote by $$u(p(\cdot )e^{-2\pi ix\cdot (\cdot )})$$. Here x is an element of $$\mathbf{R}^{n}$$ but the same assertion is valid if $$x = (x_{1},\ldots,x_{n}) \in \mathbf{R}^{n}$$ is replaced by $$z = (z_{1},\ldots,z_{n}) \in \mathbf{C}^{n}$$. In this case we define the dot product of ξ and z via $$\xi \cdot z =\sum _{ k=1}^{n}\xi _{k}z_{k}$$.

It is straightforward to verify that the function of $$z = (z_{1},\ldots,z_{n})$$

defined on $$\mathbf{C}^{n}$$ is holomorphic, in fact entire. Indeed, the continuity and linearity of u and the fact that $$(e^{-2\pi i\xi _{j}h} - 1)/h \rightarrow -2\pi i\xi _{j}$$ in $$\mathcal{C}^{\infty }(\mathbf{R}^{n})$$ as $$h \rightarrow 0$$, $$h \in \mathbf{C}$$, imply that F is holomorphic in every variable and its derivative with respect to z j is the action of the distribution u to the $$\mathcal{C}^{\infty }$$ function

By induction it follows that for all multi-indices $$\alpha$$ we have

Since F is entire, its restriction on $$\mathbf{R}^{n}$$, i.e., $$F(x_{1},\ldots,x_{n})$$, where $$x_{j} = \mbox{ Re}\,\,z_{j}$$, is real analytic. Also, an easy calculation using (2.3.4) and Leibniz’s rule yield that the restriction of $$F$$ on $$\mathbf{R}^{n}$$ and all of its derivatives have polynomial growth at infinity.

Now for f in $$\mathcal{S}(\mathbf{R}^{n})$$ we have

provided we can justify the passage of u inside the integral. The reason for this is that the Riemann sums of the integral of $$f(x)e^{-2\pi ix\cdot \xi }$$ over $$\mathbf{R}^{n}$$ converge to it in the topology of $$\mathcal{C}^{\infty }$$, and thus the linear functional u can be interchanged with the integral. We conclude that the tempered distribution $$\widehat{u}$$ can be identified with the real analytic function $$x\mapsto F(x)$$ whose derivatives have polynomial growth at infinity.

To justify the fact concerning the convergence of the Riemann sums, we argue as in the proof of the previous theorem. For each $$N = 1,2,\ldots$$, consider a partition of [−N, N] n into $$(2N^{2})^{n}$$ cubes Q m of side length 1∕N and let y m be the center of each $$Q_{m}$$. For a multi-index $$\alpha$$ let

We must show that for every M > 0, $$\sup _{\vert \xi \vert \leq M}\vert D_{N}(\xi )\vert $$ converges to zero as $$N \rightarrow \infty $$. Setting $$g(x) = f(x)(-2\pi ix)^{\alpha }$$, we write

Using the mean value theorem, we bound the absolute value of the expression inside the square brackets by

for some point z m in the cube Q m . Since

for | ξ | ≤ M, it follows that $$\sup _{\vert \xi \vert \leq M}\vert D_{N}(\xi )\vert \rightarrow 0$$ as $$N \rightarrow \infty $$. □ 

Next we give a proposition that extends the properties of the Fourier transform to tempered distributions.

Proposition 2.3.22.

Given u, v in $$\mathcal{S}'(\mathbf{R}^{n})$$ , $$f_{j},f \in \mathcal{S}$$ , $$y \in \mathbf{R}^{n}$$ , b a complex scalar, $$\alpha$$ a multi-index, and a > 0, we have

(1)

$$\widehat{u + v} =\widehat{ u} +\widehat{ v}\!$$ ,

(2)

$$\widehat{bu} = b\widehat{u}\!$$ ,

(3)

If $$f_{j} \rightarrow f$$ in $$\mathcal{S}$$ , then $$\widehat{f_{j}} \rightarrow \widehat{ f}$$ in $$\mathcal{S}$$ and if $$u_{j} \rightarrow u$$ in $$\mathcal{S}'$$ , then $$\widehat{u}_{j} \rightarrow \widehat{ u}$$ in $$\mathcal{S}'\!$$ ,

(4)

$$(\widetilde{u}\,)\,\,\widehat{\,} = (\widehat{u}\,)\,\,\widetilde{}\!$$ ,

(5)

$$(\tau ^{y}u)\,\,\widehat{\,} = e^{-2\pi iy\cdot \xi }\widehat{u}\!$$ ,

(6)

$$(e^{2\pi ix\cdot y}u)\,\,\widehat{\,} =\tau ^{y}\widehat{u}$$ ,

(7)

$$(\delta ^{a}u)\,\,\widehat{\,} = (\widehat{u}\,)_{a} = a^{-n}\delta ^{a^{-1} }\widehat{u}$$ ,

(8)

$$(\partial ^{\alpha }u)\,\,\widehat{\,} = (2\pi i\xi )^{\alpha }\widehat{u}\!$$ ,

(9)

$$\partial ^{\alpha }\widehat{u} = ((-2\pi ix)^{\alpha }u)\,\,\widehat{\,}\!$$ ,

(10)

$$(\widehat{u}\,)\,^{\vee }\, = u\!$$ ,

(11)

$$\widehat{f {\ast} u} =\widehat{ f}\,\,\widehat{u}\!$$ ,

(12)

$$\widehat{fu} =\widehat{ f} {\ast}\widehat{ u}\!$$ ,

(13)

(Leibniz’s rule) $$\partial _{j}^{m}(fu) =\sum _{ k=0}^{m}\binom{m}{k}(\partial _{j}^{k}f)(\partial _{j}^{m-k}u)$$ , $$m \in \mathbf{Z}^{+}\!$$ ,

(14)

(Leibniz’s rule) ,

(15)

If $$u_{k}$$ , $$u \in L^{p}(\mathbf{R}^{n})$$ and $$u_{k} \rightarrow u$$ in $$L^{p}$$ ( $$1 \leq p \leq \infty $$ ), then $$u_{k} \rightarrow u$$ in $$\mathcal{S}'(\mathbf{R}^{n})$$ . Therefore, convergence in $$\mathcal{S}$$ implies convergence in $$L^{p}$$ , which in turn implies convergence in $$\mathcal{S}'(\mathbf{R}^{n})$$ .

Proof.

All the statements can be proved easily using duality and the corresponding statements for Schwartz functions. □ 

We continue with an application of Theorem 2.3.21.

Proposition 2.3.23.

Given $$u \in \mathcal{S}'(\mathbf{R}^{n})$$ , there exists a sequence of $$\mathcal{C}_{0}^{\infty }$$ functions f k such that f k → u in the sense of tempered distributions; in particular, $$\mathcal{C}_{0}^{\infty }(\mathbf{R}^{n})$$ is dense in $$\mathcal{S}'(\mathbf{R}^{n})$$ .

Proof.

Fix a function in $$\mathcal{C}_{0}^{\infty }(\mathbf{R}^{n})$$ with $$\varphi (x) = 1$$ in a neighborhood of the origin. Let $$\varphi _{k}(x) =\delta ^{1/k}(\varphi )(x) =\varphi (x/k)$$. It follows from Exercise 2.3.5 (b) that for $$u \in \mathcal{S}'(\mathbf{R}^{n})$$, $$\varphi _{k}u \rightarrow u$$ in $$\mathcal{S}'$$. By Proposition 2.3.22 (3), we have that the map $$u\mapsto (\varphi _{k}\widehat{u}\,)\,^{\vee }\,\,$$ is continuous on $$\mathcal{S}'(\mathbf{R}^{n})$$. Now Theorem 2.3.21 gives that $$(\varphi _{k}\widehat{u}\,)\,^{\vee }\,$$ is a $$\mathcal{C}^{\infty }$$ function and therefore $$\varphi _{j}(\varphi _{k}\widehat{u}\,)\,^{\vee }\,$$ is in $$\mathcal{C}_{0}^{\infty }(\mathbf{R}^{n})$$. As observed, $$\varphi _{j}(\varphi _{k}\widehat{u}\,)\,^{\vee }\,\rightarrow (\varphi _{k}\widehat{u})\,^{\vee }\,$$ in $$\mathcal{S}'$$ when $$k$$ is fixed and $$j \rightarrow \infty $$. Exercise 2.3.5 (c) gives that the diagonal sequence $$\varphi _{k}(\varphi _{k}f)^{\wedge }\,$$ converges to $$\widehat{f}$$ in $$\mathcal{S}$$ as $$k \rightarrow \infty $$ for all $$f \in \mathcal{S}$$. Using duality and Exercise 2.2.2, we conclude that the sequence of $$\mathcal{C}_{0}^{\infty }$$ functions $$\varphi _{k}(\varphi _{k}\widehat{u}\,)\,^{\vee }\,$$ converges to u in $$\mathcal{S}'$$ as $$k \rightarrow \infty $$. □ 

Exercises

2.3.1. Show that a positive measure μ that satisfies

for some k > 0, can be identified with a tempered distribution. Show that if we think of Lebesgue measure as a tempered distribution, then it coincides with the constant function 1 also interpreted as a tempered distribution.

2.3.2. Let $$\varphi,f \in \mathcal{S}(\mathbf{R}^{n}),$$ and for $$\varepsilon > 0$$ let $$\varphi _{\varepsilon }(x) =\varepsilon ^{-n}\varphi (\varepsilon ^{-1}x).$$ Prove that $$\varphi _{\varepsilon } {\ast} f \rightarrow b\,f$$ in $$\mathcal{S}$$, where b is the integral of $$\varphi$$.

2.3.3. Prove that for all a > 0, $$u \in \mathcal{S}'(\mathbf{R}^{n})$$, and $$f \in \mathcal{S}(\mathbf{R}^{n})$$ we have

2.3.4. (a) Prove that the derivative of $$\chi _{[a,b]}$$ is $$\delta _{a} -\delta _{b}$$.

(b) Compute $$\partial _{j}\chi _{B(0,1)}$$ on $$\mathbf{R}^{2}$$.

(c) Compute the Fourier transforms of the locally integrable functions sinx and cosx.

(d) Prove that the derivative of the distribution $$\log \vert x\vert \in \mathcal{S}'(\mathbf{R})$$ is the distribution

2.3.5.

Let $$f \in \mathcal{S}(\mathbf{R}^{n})$$ and let $$\varphi \in \mathcal{C}_{0}^{\infty }$$ be identically equal to $$1$$ in a neighborhood of the origin. Define $$\varphi _{k}(x) =\varphi (x/k)$$ as in the proof of Proposition 2.3.23.

(a) Prove that $$(\tau ^{-he_{j}}f - f)/h \rightarrow \partial _{j}f$$ in $$\mathcal{S}$$ as h → 0. 

(b) Prove that $$\varphi _{k}f \rightarrow f$$ in $$\mathcal{S}$$ as $$k \rightarrow \infty $$.

(c) Prove that the sequence $$\varphi _{k}(\varphi _{k}f)^{\wedge }$$ converges to $$\widehat{f}$$ in $$\mathcal{S}$$ as $$k \rightarrow \infty $$.

2.3.6.

Use Theorem 2.3.21 to show that there does not exist a nonzero $$\mathcal{C}_{0}^{\infty }$$ function whose Fourier transform is also a $$\mathcal{C}_{0}^{\infty }$$ function.

2.3.7.

Let $$f \in L^{p}(\mathbf{R}^{n})$$ for some $$1 \leq p < \infty $$. Show that the sequence of functions

converges to $$\widehat{f}$$ in $$\mathcal{S}'$$.

2.3.8.

Let $$(c_{k})_{k\in \mathbf{Z}^{n}}$$ be a sequence that satisfies $$\vert c_{k}\vert \leq A(1 + \vert k\vert )^{M}$$ for all k and some fixed M and A > 0. Let $$\delta _{k}$$ denote Dirac mass at the integer k. Show that the sequence of distributions

converges to some tempered distribution u in $$\mathcal{S}'(\mathbf{R}^{n})$$ as $$N \rightarrow \infty $$. Also show that $$\widehat{u}$$ is the $$\mathcal{S}'$$ limit of the sequence of functions

2.3.9.

A distribution in $$\mathcal{S}'(\mathbf{R}^{n})$$ is called homogeneous of degree $$\gamma \in \mathbf{C}$$ if for all $$\lambda > 0$$ and for all $$\varphi \in \mathcal{S}(\mathbf{R}^{n})$$ we have

(a) Prove that this definition agrees with the usual definition for functions.

(b) Show that $$\delta _{0}$$ is homogeneous of degree − n.

(c) Prove that if u is homogeneous of degree $$\gamma$$, then $$\partial ^{\alpha }u$$ is homogeneous of degree $$\gamma -\vert \alpha \vert $$.

(d) Show that u is homogeneous of degree $$\gamma$$ if and only if $$\widehat{u}$$ is homogeneous of degree $$-n-\gamma$$.

2.3.10. (a) Show that the functions $$e^{inx}$$ and $$e^{-inx}$$ converge to zero in $$\mathcal{S}'$$ and $$\mathcal{D}'$$ as $$n \rightarrow \infty $$. Conclude that multiplication of distributions is not a continuous operation even when it is defined. (b) What is the limit of $$\sqrt{n}(1 + n\vert x\vert ^{2})^{-1}$$ in $$\mathcal{D}'(\mathbf{R})$$ as $$n \rightarrow \infty $$?

2.3.11.

(S. Bernstein) Let f be a bounded function on $$\mathbf{R}^{n}$$ with $$\widehat{f}$$ supported in the ball $$B(0,R)$$. Prove that for all multi-indices $$\alpha$$ there exist constants $$C_{\alpha,n}$$ (depending only on $$\alpha$$ and on the dimension n) such that

[Hint: Write $$f = f {\ast} h_{1/R}$$, where h is a Schwartz function h in $$\mathbf{R}^{n}$$ whose Fourier transform is equal to one on the ball B(0, 1) and vanishes outside the ball B(0, 2).]

2.3.12. Let $$\widehat{\varPhi }$$ be a $$\mathcal{C}_{0}^{\infty }$$ function that is equal to 1 in B(0, 1) and let $$\widehat{\varTheta }$$ be a $$\mathcal{C}^{\infty }$$ function that is equal to 1 in a neighborhood of infinity and equal to zero in a neighborhood of the origin. Prove the following.(a) For all u in $$\mathcal{S}'(\mathbf{R}^{n})$$ we have

(b) For all u in $$\mathcal{S}'(\mathbf{R}^{n})$$ we have

2.3.13. Prove that there exists a function in $$L^{p}$$ for $$2 < p < \infty $$ whose distributional Fourier transform is not a locally integrable function.

[Hint: Assume the converse. Then for all $$f \in L^{p}(\mathbf{R}^{n})$$, $$\widehat{f}$$ is locally integrable and hence the map $$f\mapsto \widehat{f}$$ is a well defined linear operator from $$L^{p}(\mathbf{R}^{n})$$ to $$L^{1}(B(0,M))$$ for all M > 0 (i.e. $$\|\widehat{f}\,\|_{L^{1}(B(0,M))} < \infty $$ for all $$f \in L^{p}(\mathbf{R}^{n})$$). Use the closed graph theorem to deduce that $$\|\widehat{f}\,\|_{L^{1}(B(0,M))} \leq C_{M}\|f\|_{L^{p}(\mathbf{R}^{n})}$$ for some $$C_{M} < \infty $$. To violate this inequality whenever p > 2, take $$f_{N}(x) = (1 + iN)^{-n/2}e^{-\pi (1+iN)^{-1}\vert x\vert ^{2} }$$ and let $$N \rightarrow \infty $$, noting that $$\widehat{f_{N}}(\xi ) = e^{-\pi \vert \xi \vert ^{2}(1+iN) }$$.]

2.4 More About Distributions and the Fourier Transform

In this section we discuss further properties of distributions and Fourier transforms and bring up certain connections that arise between harmonic analysis and partial differential equations.

2.4.1 Distributions Supported at a Point

We begin with the following characterization of distributions supported at a single point.

Proposition 2.4.1.

If $$u \in \mathcal{S}'(\mathbf{R}^{n})$$ is supported in the singleton $$\{x_{0}\}$$ , then there exists an integer $$k$$ and complex numbers $$a_{\alpha }$$ such that

Proof.

Without loss of generality we may assume that x ₀ = 0. By (2.3.3) we have that for some C, m, and k,

We now prove that if $$\varphi \in \mathcal{S}$$ satisfies

(2.4.1)

then $$\big\langle u,\varphi \big\rangle = 0$$. To see this, fix a $$\varphi$$ satisfying (2.4.1) and let $$\zeta (x)$$ be a smooth function on $$\mathbf{R}^{n}$$ that is equal to 1 when | x | ≥ 2 and equal to zero for | x | ≤ 1. Let $$\zeta ^{\varepsilon }(x) =\zeta (x/\varepsilon )$$. Then, using (2.4.1) and the continuity of the derivatives of $$\varphi$$ at the origin, it is not hard to show that $$\rho _{\alpha,\beta }(\zeta ^{\varepsilon }\varphi -\varphi ) \rightarrow 0$$ as $$\varepsilon \rightarrow 0$$ for all $$\vert \alpha \vert \leq m$$ and $$\vert \beta \vert \leq k$$. Then

as $$\varepsilon \rightarrow 0$$. This proves our assertion.

Now let $$f \in \mathcal{S}(\mathbf{R}^{n})$$. Let η be a $$\mathcal{C}_{0}^{\infty }$$ function on $$\mathbf{R}^{n}$$ that is equal to 1 in a neighborhood of the origin. Write

(2.4.2)

where $$h(x) = O(x^{k+1})$$ as | x | → 0. Then η h satisfies (2.4.1) and hence $$\big\langle u,\eta h\big\rangle = 0$$ by the claim. Also,

by our hypothesis. Applying u to both sides of (2.4.2), we obtain

with $$a_{\alpha } = (-1)^{\vert \alpha \vert }u(x^{\alpha }\eta (x))/\alpha !$$. This proves the proposition. □ 

An immediate consequence is the following result.

Corollary 2.4.2.

Let $$u \in \mathcal{S}'(\mathbf{R}^{n})$$ . If $$\widehat{u}$$ is supported in the singleton {ξ ₀ }, then u is a finite linear combination of functions $$(-2\pi i\xi )^{\alpha }e^{2\pi i\xi \cdot \xi _{0}}$$ , where $$\alpha$$ is a multi-index. In particular, if $$\widehat{u}$$ is supported at the origin, then u is a polynomial.

Proof.

Proposition 2.4.1 gives that $$\widehat{u}$$ is a linear combination of derivatives of Dirac masses at $$\xi _{0}$$. Then Proposition 2.3.22 (8) yields the required conclusion. □ 

2.4.2 The Laplacian

The Laplacian $$\varDelta$$ is a partial differential operator acting on tempered distributions on $$\mathbf{R}^{n}$$ as follows:

Solutions of Laplace’s equation $$\varDelta (u) = 0$$ are called harmonic distributions. We have the following:

Corollary 2.4.3.

Let $$u \in \mathcal{S}'(\mathbf{R}^{n})$$ satisfy $$\varDelta (u) = 0$$ . Then u is a polynomial.

Proof.

Taking Fourier transforms, we obtain that $$\widehat{\varDelta (u)} = 0$$. Therefore,

This implies that $$\widehat{u}$$ is supported at the origin, and by Corollary 2.4.2 it follows that u must be polynomial. □ 

Liouville’s classical theorem that every bounded harmonic function must be constant is a consequence of Corollary 2.4.3. See Exercise 2.4.2.

Next we would like to compute the fundamental solutions of Laplace’s equation in $$\mathbf{R}^{n}$$. A distribution is called a fundamental solution of a partial differential operator L if we have $$L(u) =\delta _{0}$$. The following result gives the fundamental solution of the Laplacian.

Proposition 2.4.4.

For $$n \geq 3$$ we have

(2.4.3)

while for n = 2,

(2.4.4)

Proof.

We use Green’s identity

where $$\varOmega$$ is an open set in $$\mathbf{R}^{n}$$ with smooth boundary and $$\partial v/\partial \nu$$ denotes the derivative of v with respect to the outer unit normal vector. Take $$\varOmega = \mathbf{R}^{n}\setminus \overline{B(0,\varepsilon )}$$, v =  | x | 2−n , and u = f a $$\mathcal{C}_{0}^{\infty }(\mathbf{R}^{n})$$ function in the previous identity. The normal derivative of $$f(r\theta )$$ is the derivative with respect to the radial variable r. Observe that $$\varDelta (\vert x\vert ^{2-n}) = 0$$ for x ≠ 0. We obtain

(2.4.5)

where dθ denotes surface measure on the sphere $$\vert \theta \vert =\varepsilon$$. Now observe two things: first, that for some C = C(f) we have

second, that

as $$\varepsilon \rightarrow 0$$. Letting $$\varepsilon \rightarrow 0$$ in (2.4.5), we obtain that

which implies (2.4.3) in view of the formula for $$\omega _{n-1}$$ given in Appendix A.3.

The proof of (2.4.4) is identical. The only difference is that the quantity $$\partial r^{2-n}/\partial r$$ in (2.4.5) is replaced by $$\partial \log r/\partial r.$$ □ 

2.4.3 Homogeneous Distributions

The fundamental solutions of the Laplacian are locally integrable functions on $$\mathbf{R}^{n}$$ and also homogeneous of degree 2 − n when n ≥ 3. Since homogeneous distributions often arise in applications, it is desirable to pursue their study. Here we do not undertake such a study in depth, but we discuss a few important examples.

Our first goal is to understand the action of the distribution | t |  z on $$\mathbf{R}^{n}$$ when $$\mbox{ Re}\,z \leq -n$$. Let us consider first the case n = 1. The tempered distribution

is well-defined when $$\mbox{ Re}\,z > -1$$. But we can extend the definition for all z with $$\mbox{ Re}\,z > -3$$ and z ≠ − 1 by rewriting it as

(2.4.6)

and noting that for all $$\varphi \in \mathcal{S}(\mathbf{R})$$ we have

thus $$w_{z} \in \mathcal{S}'(\mathbf{R})$$. Subtracting the Taylor polynomial of degree 3 centered at zero from $$\varphi (t)$$ instead of the linear one, as in (2.4.6), allows us to extend the definition for $$\mbox{ Re}\,z > -5$$ and $$\mbox{ Re}\,z\notin \{ - 1,-3\}$$. Subtracting higher order Taylor polynomials allows us to extend the definition of w z for all $$z \in \mathbf{C}$$ except at the negative odd integers. To be able to include the points $$z = -1,-3,-5,-7,\ldots$$ we need to multiply w z by an entire function that has simple zeros at all the negative odd integers to be able to eliminate the simple poles at these points. Such a function is $$\varGamma \big(\tfrac{z+1} {2} \big)^{-1}$$. This discussion leads to the following definition.

Definition 2.4.5.

For $$z \in \mathbf{C}$$ we define a distribution u z as follows:

(2.4.7)

Clearly the $$u_{z}$$’s coincide with the locally integrable functions

when Re z > −n and the definition makes sense only for that range of z’s. It follows from its definition that u z is a homogeneous distribution of degree z.

We would like to extend the definition of u z for $$z \in \mathbf{C}$$. Let $$\mbox{ Re}\,\,z > -n$$ first. Fix N to be a positive integer. Given $$f \in \mathcal{S}(\mathbf{R}^{n})$$, write the integral in (2.4.7) as follows:

The preceding expression is equal to

where we switched to polar coordinates in the penultimate integral. Now set

where $$\alpha = (\alpha _{1},\ldots,\alpha _{n})$$ is a multi-index. These coefficients are zero when at least one $$\alpha _{j}$$ is odd. Consider now the case that all the $$\alpha _{j}$$’s are even; then $$\vert \alpha \vert $$ is also even. The function $$\varGamma (\frac{z+n} {2} )$$ has simple poles at the points

see Appendix A.5. These poles cancel exactly the poles of the function

at $$z = -n -\vert \alpha \vert $$ when $$\vert \alpha \vert $$ is an even integer in [0, N]. We therefore have

(2.4.8)

Both integrals converge absolutely when $$\mbox{ Re}\,\,z > -N - n - 1$$, since the expression inside the curly brackets above is bounded by a constant multiple of | x |  N+1, and the resulting function of z in (2.4.8) is a well defined analytic function in the range $$\mbox{ Re}\,\,z > -N - n - 1$$.

Since N was arbitrary, $$\langle u_{z},f\rangle$$ has an analytic extension to all of $$\mathbf{C}$$. Therefore, u z is a distribution-valued entire function of z, i.e., for all $$\varphi \in \mathcal{S}(\mathbf{R}^{n})$$, the function $$z\mapsto \langle u_{z},\varphi \rangle$$ is entire.

Next we would like to calculate the Fourier transform of u z . We know by Exercise 2.3.9 that $$\widehat{u_{z}}$$ is a homogeneous distribution of degree − n − z. The choice of constant in the definition of u z was made to justify the following result:

Theorem 2.4.6.

For all $$z \in \mathbf{C}$$ we have $$\widehat{u_{z}} = u_{-n-z}$$ .

Proof.

The idea of the proof is straightforward. First we show that for a certain range of z’s we have

(2.4.9)

for some fixed constant C(n, z) and all $$\varphi \in \mathcal{S}(\mathbf{R}^{n})$$. Next we pick a specific $$\varphi$$ to evaluate the constant C(n, z). Then we use analytic continuation to extend the validity of (2.4.9) for all z’s. Use polar coordinates by setting $$\xi =\rho \varphi$$ and x = rθ in (2.4.9). We have

where we set

(2.4.10)

(2.4.11)

and the second equality in (2.4.10) is a consequence of rotational invariance. It remains to prove that the integral in (2.4.11) converges for some range of z’s.

If n = 1, then

and the integral in (2.4.11) converges conditionally for $$-1 < \mbox{ Re}\,\,z < 0$$.

Let us therefore assume that n ≥ 2. Since $$\vert \sigma _{n}(t)\vert \leq \omega _{n-1}$$, the integral converges near zero when $$-n < \mbox{ Re}\,\,z$$. Let us study the behavior of $$\sigma _{n}(t)$$ for t large. Using the formula in Appendix D.2 and the definition of Bessel functions in Appendix B.1, we write

for some constant c n . Since n ≥ 2 we have when n − 2 > −1∕2. Then the asymptotics for Bessel functions (Appendix B.7) apply and yield $$\vert \sigma _{n}(t)\vert \leq c\,t^{-(n-1)/2}$$ for t ≥ 1. Splitting the integral in (2.4.11) in t ≤ 1 and t ≥ 1 and using the corresponding estimates, we notice that it converges absolutely on [0, 1] when $$\mbox{ Re}\,z > -n$$ and on $$[1,\infty )$$ when $$\mbox{ Re}\,\,z + n - 1 -\frac{n-1} {2} < -1$$.

We have now proved that when $$-n < \mbox{ Re}\,\,z < -\frac{n+1} {2}$$ and n ≥ 2 we have

for some constant C(n, z) that we wish to compute. Insert the function $$\varphi (x) = e^{-\pi \vert x\vert ^{2} }$$ in (2.4.9). Example 2.2.9 gives that this function is equal to its Fourier transform. Use polar coordinates to write

Change variables $$s =\pi r^{2}$$ and use the definition of the gamma function to obtain that

It follows that $$\widehat{u_{z}} = u_{-n-z}$$ for the range of z’s considered.

At this point observe that for every $$f \in \mathcal{S}(\mathbf{R}^{n})$$, the function $$z\mapsto \langle \widehat{u_{z}} - u_{-z-n},f\rangle$$ is entire and vanishes for $$-n < \mbox{ Re}\,z < -n + 1/2$$. Therefore, it must vanish everywhere and the theorem is proved. □ 

Homogeneous distributions were introduced in Exercise 2.3.9. We already saw that the Dirac mass on $$\mathbf{R}^{n}$$ is a homogeneous distribution of degree − n. There is another important example of a homogeneous distributions of degree − n, which we now discuss.

Let $$\varOmega$$ be an integrable function on the sphere $$\mathbf{S}^{n-1}$$ with integral zero. Define a tempered distribution $$W_{\varOmega }$$ on $$\mathbf{R}^{n}$$ by setting

(2.4.12)

We check that $$W_{\varOmega }$$ is a well defined tempered distribution on $$\mathbf{R}^{n}$$. Indeed, since $$\varOmega (x/\vert x\vert )/\vert x\vert ^{n}$$ has integral zero over all annuli centered at the origin, we obtain

for suitable constants C ₁ and C ₂ in view of (2.2.2).

One can verify that $$W_{\varOmega } \in \mathcal{S}'(\mathbf{R}^{n})$$ is a homogeneous distribution of degree − n just like the Dirac mass at the origin. It is an interesting fact that all homogeneous distributions on $$\mathbf{R}^{n}$$ of degree − n that coincide with a smooth function away from the origin arise in this way. We have the following result.

Proposition 2.4.7.

Suppose that m is a $$\mathcal{C}^{\infty }$$ function on $$\mathbf{R}^{n}\setminus \{0\}$$ that is homogeneous of degree zero. Then there exist a scalar b and a $$\mathcal{C}^{\infty }$$ function $$\varOmega$$ on $$\mathbf{S}^{n-1}$$ with integral zero such that

(2.4.13)

where $$W_{\varOmega }$$ denotes the distribution defined in(2.4.12).

To prove this result we need the following proposition, whose proof we postpone until the end of this section.

Proposition 2.4.8.

Suppose that u is a $$\mathcal{C}^{\infty }$$ function on $$\mathbf{R}^{n}\setminus \{0\}$$ that is homogeneous of degree $$z \in \mathbf{C}$$ . Then $$\widehat{u}$$ is a $$\mathcal{C}^{\infty }$$ function on $$\mathbf{R}^{n}\setminus \{0\}$$ .

We now prove Proposition 2.4.7 using Proposition 2.4.8.

Proof.

Let a be the integral of the smooth function $$m$$ over $$\mathbf{S}^{n-1}$$. The function m − a is homogeneous of degree zero and thus locally integrable on $$\mathbf{R}^{n}$$; hence it can be thought of as a tempered distribution that we call $$\widehat{u}$$ (the Fourier transform of a tempered distribution u). Since $$\widehat{u}$$ is a $$\mathcal{C}^{\infty }$$ function on $$\mathbf{R}^{n}\setminus \{0\}$$, Proposition 2.4.8 implies that $$u$$ is also a $$\mathcal{C}^{\infty }$$ function on $$\mathbf{R}^{n}\setminus \{0\}$$. Let $$\varOmega$$ be the restriction of u on $$\mathbf{S}^{n-1}$$. Then $$\varOmega$$ is a well defined $$\mathcal{C}^{\infty }$$ function on $$\mathbf{S}^{n-1}$$. Since u is a homogeneous function of degree − n that coincides with the smooth function $$\varOmega$$ on $$\mathbf{S}^{n-1}$$, it follows that $$u(x) =\varOmega (x/\vert x\vert )/\vert x\vert ^{n}$$ for $$x$$ in $$\mathbf{R}^{n}\setminus \{0\}$$.

We show that $$\varOmega$$ has mean value zero over $$\mathbf{S}^{n-1}$$. Pick a nonnegative, radial, smooth, and nonzero function ψ on $$\mathbf{R}^{n}$$ supported in the annulus 1 <  | x |  < 2. Switching to polar coordinates, we write

and thus $$\varOmega$$ has mean value zero over $$\mathbf{S}^{n-1}$$ (since c ψ ≠ 0).

We can now legitimately define the distribution $$W_{\varOmega }$$, which coincides with the function $$\varOmega (x/\vert x\vert )/\vert x\vert ^{n}$$ on $$\mathbf{R}^{n}\setminus \{0\}$$. But the distribution u also coincides with this function on $$\mathbf{R}^{n}\setminus \{0\}$$. It follows that $$u - W_{\varOmega }$$ is supported at the origin. Proposition 2.4.1 now gives that $$u - W_{\varOmega }$$ is a sum of derivatives of Dirac masses. Since both distributions are homogeneous of degree − n, it follows that

But $$u = (m - a)\!\,^{\vee }\,\, = m\!\,^{\vee }\,- a\delta _{0}$$, and thus $$m\!\,^{\vee }\, = (c + a)\delta _{0} + W_{\varOmega }.$$ This proves the proposition. □ 

We now turn to the proof of Proposition 2.4.8.

Proof.

Let $$u \in \mathcal{S}'$$ be homogeneous of degree $$z$$ and $$\mathcal{C}^{\infty }$$ on $$\mathbf{R}^{n}\setminus \{0\}$$. We need to show that $$\widehat{u}$$ is $$\mathcal{C}^{\infty }$$ away from the origin. We prove that $$\widehat{u}$$ is $$\mathcal{C}^{M}$$ for all $$M$$. Fix $$M \in \mathbf{Z}^{+}$$ and let $$\alpha$$ be any multi-index such that

(2.4.14)

Pick a $$\mathcal{C}^{\infty }$$ function $$\varphi$$ on $$\mathbf{R}^{n}$$ that is equal to 1 when | x | ≥ 2 and equal to zero for | x | ≤ 1. Write $$u_{0} = (1-\varphi )u$$ and $$u_{\infty } =\varphi u$$. Then

where the operations are performed in the sense of distributions. Since u ₀ is compactly supported, Theorem 2.3.21 implies that $$\widehat{\partial ^{\alpha }u_{0}}$$ is $$\mathcal{C}^{\infty }$$. Now Leibniz’s rule gives that

where v is a smooth function supported in the annulus $$1 \leq \vert x\vert \leq 2$$. Then $$\widehat{v}$$ is $$\mathcal{C}^{\infty }$$ and we need to show only that $$\widehat{\varphi \partial ^{\alpha }u}$$ is $$\mathcal{C}^{M}$$. The function $$\varphi \partial ^{\alpha }u$$ is actually $$\mathcal{C}^{\infty }$$, and by the homogeneity of $$\partial ^{\alpha }u$$ (Exercise 2.3.9 (c)) we obtain that $$(\partial ^{\alpha }u)(x) = \vert x\vert ^{-\vert \alpha \vert +z}(\partial ^{\alpha }u)(x/\vert x\vert )$$. Since $$\varphi$$ is supported away from zero, it follows that

(2.4.15)

for some $$C_{\alpha } > 0$$. It is now straightforward to see that if a function satisfies (2.4.15), then its Fourier transform is $$\mathcal{C}^{M}$$ whenever (2.4.14) is satisfied. See Exercise 2.4.1.

We conclude that $$\widehat{\partial ^{\alpha }u_{\infty }}$$ is a $$\mathcal{C}^{M}$$ function whenever (2.4.14) is satisfied; thus so is $$\widehat{\partial ^{\alpha }u}$$. Since $$\widehat{\partial ^{\alpha }u}(\xi ) = (2\pi i\xi )^{\alpha }\widehat{u}(\xi )$$, we deduce smoothness for $$\widehat{u}$$ away from the origin. Let ξ ≠ 0. Pick a neighborhood V of ξ such that for η in V we have $$\eta _{j}\neq 0$$ for some $$j \in \{ 1,\ldots,n\}$$. Consider the multi-index $$(0,\ldots,\vert \alpha \vert,\ldots,0)$$ with $$\vert \alpha \vert $$ in the jth coordinate and zeros elsewhere. Then $$(2\pi i\eta _{j})^{\vert \alpha \vert }\widehat{u}(\eta )$$ is a $$\mathcal{C}^{M}$$ function on V, and thus so is $$\widehat{u}(\eta )$$, since we can divide by $$\eta _{j}^{\vert \alpha \vert }$$. We conclude that $$\widehat{u}(\xi )$$ is $$\mathcal{C}^{M}$$ on $$\mathbf{R}^{n}\setminus \{0\}$$. Since M is arbitrary, the conclusion follows. □ 

We end this section with an example that illustrates the usefulness of some of the ideas discussed in this section.

Example 2.4.9.

Let η be a smooth radial function on $$\mathbf{R}^{n}$$ that is equal to 1 on the set | x | ≥ 1∕2 and vanishes on the set | x | ≤ 1∕4. Fix $$z \in \mathbf{C}$$ satisfy $$0 < \mbox{ Re}\,z < n$$. Let $$g =\big (\eta (x)\vert x\vert ^{-z}\big)^{\wedge }$$ be the distributional Fourier transform of $$\eta (x)\vert x\vert ^{-z}$$. We show that g is a function that decays faster than | ξ | −N at infinity (for sufficiently large positive number N) and that

(2.4.16)

is a $$\mathcal{C}^{\infty }$$ function on $$\mathbf{R}^{n}$$. This example indicates the interplay between the smoothness of a function and the decay of its Fourier transform. The smoothness of the function $$\eta (x)\vert x\vert ^{-z}$$ near zero has as a consequence the rapid decay of g near infinity, while the slow decay of η(x) | x | −z at infinity reflects the lack of smoothness of g(ξ) at zero, in view of the moderate blowup $$\vert \xi \vert ^{\mbox{ Re}\,z-n}$$ as $$\vert \xi \vert \rightarrow 0$$.

To show that g is a function we write it as $$g = (\vert x\vert ^{-z})^{\wedge } +\big ((\eta (x) - 1)\vert x\vert ^{-z}\big)^{\wedge }$$ and we observe that the first term is a function, since $$0 < \mbox{ Re}\,z < n$$. Using Theorem 2.4.6 we write

where $$\widehat{\varphi }(\xi ) =\big ((\eta (x) - 1)\vert x\vert ^{-z}\big)^{\wedge }(\xi )$$ is a $$\mathcal{C}^{\infty }$$ function, since it is the Fourier transform of a compactly supported integrable function. This proves that g is a function and that the difference in (2.4.16) is $$\mathcal{C}^{\infty }$$.

Finally, we assert that every derivative of g satisfies $$\vert \partial ^{\gamma }g(\xi )\vert \leq C_{\gamma,N}\vert \xi \vert ^{-N}$$ for all sufficiently large positive integers N when ξ ≠ 0. Indeed, fix a multi-index $$\gamma$$ and write $$\partial ^{\gamma }g(\xi ) = (\vert x\vert ^{-z}\eta (x)(-2\pi ix)^{\gamma })^{\wedge }(\xi )$$. It follows that

for all $$N \in \mathbf{Z}^{+}$$, where $$\varDelta$$ is the Laplacian in the x variable. Using Leibniz’s rule we distribute $$\varDelta ^{N}$$ to the product. If a derivative falls on η, we obtain a compactly supported smooth function, hence integrable. If all derivatives fall on $$\vert x\vert ^{-z}x^{\gamma }$$, then we obtain a term that decays like $$\vert x\vert ^{-\mbox{ Re}\,z+\vert \gamma \vert -2N}$$ at infinity, which is also integrable if $$N$$ is sufficiently large. Thus the function $$\vert \xi \vert ^{2N}\vert \partial ^{\gamma }g(\xi )\vert $$ is equal to the Fourier transform of an L ¹ function, hence it is bounded, when $$2N > n -\mbox{ Re}\,z + \vert \gamma \vert $$.

Exercises

2.4.1. Suppose that a function f satisfies the estimate

for some C > 0 and N > n + 1. Then $$\widehat{f}$$ is $$\mathcal{C}^{M}$$ for all $$M \in \mathbf{Z}^{+}$$ with 1 ≤ M < N − n.

2.4.2. Use Corollary 2.4.3 to prove Liouville’s theorem that every bounded harmonic function on $$\mathbf{R}^{n}$$ must be a constant. Derive as a consequence the fundamental theorem of algebra , stating that every polynomial on $$\mathbf{C}$$ must have a complex root.

2.4.3. Prove that e x is not in $$\mathcal{S}'(\mathbf{R})$$ but that $$e^{x}e^{ie^{x} }$$ is in $$\mathcal{S}'(\mathbf{R})$$.

2.4.4. Show that the Schwartz function $$x\mapsto \mbox{ sech}\,(\pi x)$$, $$x \in \mathbf{R}$$, coincides with its Fourier transform.

[Hint: Integrate the function $$e^{iaz}$$ over the rectangular contour with corners (−R, 0), (R, 0), $$(R,i\pi )$$, and $$(-R,i\pi )$$.]

2.4.5. ([174]) Construct an uncountable family of linearly independent Schwartz functions f a such that $$\vert f_{a}\vert = \vert f_{b}\vert $$ and $$\vert \widehat{f_{a}}\vert = \vert \widehat{f_{b}}\vert $$ for all $$f_{a}$$ and f b in the family.

[Hint: Let w be a smooth nonzero function whose Fourier transform is supported in the interval [−1∕2, 1∕2] and let ϕ be a real-valued smooth nonconstant periodic function with period 1. Then take $$f_{a}(x) = w(x)e^{i\phi (x-a)}$$ for $$a \in \mathbf{R}$$.]

2.4.6.

Let P y be the Poisson kernel defined in (2.1.13). Prove that for $$f \in L^{p}(\mathbf{R}^{n})$$, $$1 \leq p < \infty $$, the function

is a harmonic function on $$\mathbf{R}_{+}^{n+1}$$. Use the Fourier transform and Exercise 2.2.11 to prove that $$(P_{y_{1}} {\ast} P_{y_{2}})(x) = P_{y_{1}+y_{2}}(x)$$ for all $$x \in \mathbf{R}^{n}$$.

2.4.7.

(a) For a fixed $$x_{0} \in \mathbf{S}^{n-1}$$, show that the function

is harmonic on $$\mathbf{R}^{n}\setminus \{x_{0}\}$$.

(b) For fixed $$x_{0} \in \mathbf{S}^{n-1}$$, prove that the family of functions $$\theta \mapsto v(rx_{0};\theta )$$, $$0 < r < 1$$, defined on the sphere satisfies

uniformly in $$x_{0}$$. The function $$v(rx_{0};\theta )$$ is called the Poisson kernel for the sphere.

(c) Show that

for all | x |  < 1.

(d) Let f be a continuous function on $$\mathbf{S}^{n-1}$$. Prove that the function

solves the Dirichlet problem Δ(u) = 0 on | x |  < 1 with boundary values u = f on $$\mathbf{S}^{n-1}$$, in the sense $$\lim _{r\uparrow 1}u(rx_{0}) = f(x_{0})$$ when $$\vert x_{0}\vert = 1$$.

[Hint: Part (c): Apply the mean value property over spheres to the harmonic function $$y\mapsto (1 -\vert x\vert ^{2}\vert y\vert ^{2})\big\vert \vert x\vert y - \frac{x} {\vert x\vert }\big\vert ^{-n}$$.]

2.4.8.

Fix $$n \in \mathbf{Z}^{+}$$ with n ≥ 2 and a real number $$\lambda$$, $$0 <\lambda < n$$. Also fix $$\eta \in \mathbf{S}^{n}$$ and $$y \in \mathbf{R}^{n}$$.

(a) Prove that

(b) Prove that

[Hint: Part (a): See Appendix D.4 Part (b): Use the stereographic projection in Appendix D.6.]

2.4.9.

Prove the following beta integral identity:

where $$0 <\alpha _{1},\alpha _{2} < n,\,\alpha _{1} +\alpha _{2} > n$$.

[Hint: Reduce to the case y = 0, interpret the integral as a convolution, and use Theorem 2.4.6.]

2.4.10. (a) Prove that if a continuous integrable function f on $$\mathbf{R}^{n}$$ ($$n \geq 2$$) is constant on the spheres $$r\,\mathbf{S}^{n-1}$$ for all r > 0, then so is its Fourier transform.

(b) If a continuous integrable function on $$\mathbf{R}^{n}$$ (n ≥ 3) is constant on all (n − 2)–dimensional spheres orthogonal to $$e_{1} = (1,0,\ldots,0)$$, then its Fourier transform has the same property.

2.4.11.

([137]) Suppose that $$0 < d_{1},d_{2},d_{3} < n$$ satisfy $$d_{1} + d_{2} + d_{3} = 2n.$$ Prove that for any distinct x, y, z ∈ R n we have the identity

[Hint: Reduce matters to the case that z = 0 and y = e ₁. Then take the Fourier transform in x and use that the function $$h(t) = \vert t - e_{1}\vert ^{-d_{3}}\vert t\vert ^{-d_{1}}$$ satisfies $$\widehat{h}(\xi ) =\widehat{ h}(A_{\xi }^{-2}\xi )$$ for all ξ ≠ 0, where A ξ is an orthogonal matrix with $$A_{\xi }e_{1} =\xi /\vert \xi \vert $$.]

2.4.12. (a) Integrate the function $$e^{iz^{2} }$$ over the contour consisting of the three pieces $$P_{1} =\{ x + i0:\,\, 0 \leq x \leq R\}$$, $$P_{2} =\{ Re^{i\theta }:\,\, 0 \leq \theta \leq \frac{\pi } {4}\}$$, and $$P_{3} =\{ r\,e^{i \frac{\pi }{ 4} }:\,\, 0 \leq r \leq R\}$$ (with the proper orientation) to obtain the Fresnel integral identity:

(b) Use the result in part (a) to show that the Fourier transform of the function $$e^{i\pi \vert x\vert ^{2} }$$ in $$\mathbf{R}^{n}$$ is equal to $$e^{i\frac{\pi n} {4} }e^{-i\pi \vert \xi \vert ^{2} }$$.

[Hint: Part (a): On P ₂ we have $$e^{-R^{2}\!\sin (2\theta ) } \leq e^{-\frac{4} {\pi } R^{2}\theta }$$, and the integral over P ₂ tends to 0. Part (b): Try first n = 1.]

2.5 Convolution Operators on $$L^{p}$$ Spaces and Multipliers

In this section we study the class of operators that commute with translations. We prove in this section that bounded operators that commute with translations must be of convolution type. Convolution operators arise in many situations, and we would like to know under what circumstances they are bounded between $$L^{p}$$ spaces.

2.5.1 Operators That Commute with Translations

Definition 2.5.1.

A vector space X of measurable functions on $$\mathbf{R}^{n}$$ is called closed under translations if for f ∈ X we have τ z (f) ∈ X for all $$z \in \mathbf{R}^{n}$$. Let X and Y be vector spaces of measurable functions on $$\mathbf{R}^{n}$$ that are closed under translations. Let also T be an operator from X to Y. We say that T commutes with translations or is translation-invariant if

for all f ∈ X and all $$y \in \mathbf{R}^{n}$$.

It is straightforward to see that convolution operators commute with translations, i.e., $$\tau ^{y}(f {\ast} g) =\tau ^{y}(f) {\ast} g$$ whenever the convolution is defined. One of the goals of this section is to prove the converse: every bounded linear operator that commutes with translations is of convolution type. We have the following:

Theorem 2.5.2.

Let $$1 \leq p,q \leq \infty $$ and suppose T is a bounded linear operator from $$L^{p}(\mathbf{R}^{n})$$ to $$L^{q}(\mathbf{R}^{n})$$ that commutes with translations. Then there exists a unique tempered distribution w such that

A very important point to make is that if $$p = \infty $$, the restriction of T on $$\mathcal{S}$$ does not uniquely determine T on the entire $$L^{\infty }$$; see Example 2.5.9 and the comments preceding it about this. The theorem is a consequence of the following two results:

Lemma 2.5.3.

Under the hypotheses of Theorem 2.5.2 and for $$f \in \mathcal{S}(\mathbf{R}^{n})$$ , the distributional derivatives of T(f) are $$L^{q}$$ functions that satisfy

(2.5.1)

Lemma 2.5.4.

Let $$1 \leq q \leq \infty $$ and let $$h \in L^{q}(\mathbf{R}^{n})$$ . If all distributional derivatives $$\partial ^{\alpha }h$$ are also in $$L^{q}$$ , then h is almost everywhere equal to a continuous function H satisfying

(2.5.2)

Proof.

Assuming Lemmas 2.5.3 and 2.5.4, we prove Theorem 2.5.2.

Given $$f \in \mathcal{S}(\mathbf{R}^{n})$$, by Lemmas 2.5.3 and 2.5.4, there is a continuous function H such that T(f) = H a.e. and such that

holds. Define a linear functional u on $$\mathcal{S}$$ by setting

This functional is well-defined, for, if there is another continuous function G such that G = T(f) a.e., then G = H a.e. and since both functions are continuous, it follows that H = G everywhere and thus H(0) = G(0).

By (2.5.1), (2.5.2), and the boundedness of T, we have

where the last estimate uses (2.2.8). This implies that u is in $$\mathcal{S}'$$. We now set $$w =\widetilde{ u}$$ and we claim that for all $$x \in \mathbf{R}^{n}$$ we have

(2.5.3)

Assuming (2.5.3) we prove that T(f) = f ∗ w for $$f \in \mathcal{S}$$. To see this, by Theorem 2.3.20 and by the translation invariance of T, for a given $$f \in \mathcal{S}(\mathbf{R}^{n})$$ we have

where the last equality holds for almost all x, by the definition of H. Thus f ∗ w = T(f) a.e., as claimed. The uniqueness of w follows from the simple observation that if f ∗ w = f ∗ w′ for all $$f \in \mathcal{S}(\mathbf{R}^{n})$$, then w = w′.

We now turn to the proof of (2.5.3). Given $$f \in \mathcal{S}(\mathbf{R}^{n})$$ and $$x \in \mathbf{R}^{n}$$ and let H x be the continuous function such that $$H_{x} = T(\tau ^{-x}f)$$. We show that $$H_{x}(0) = H(x)$$. Indeed, we have

where the equality $$T(f)(x + y) = H(x + y)$$ holds a.e. in y. Thus the continuous functions H x and $$\tau ^{-x}H$$ are equal a.e. and thus they must be everywhere equal, in particular, when y = 0. This proves that H x (0) = H(x), which is a restatement of (2.5.3). □ 

We now return to Lemmas 2.5.3 and 2.5.4. We begin with Lemma 2.5.3.

Proof.

Consider first the multi-index $$\alpha = (0,\ldots,1,\ldots,0)$$, where 1 is in the jth entry and 0 is elsewhere. Let $$e_{j} = (0,\ldots,1,\ldots,0)$$, where 1 is in the jth entry and zero elsewhere. We have

(2.5.4)

since both of these expressions are equal to

and T commutes with translations. We will let h → 0 in both sides of (2.5.4). We write

from which it follows that for | h |  < 1∕2 we have

The integrand on the left-hand side of (2.5.4) is bounded by the integrable function $$\vert T(f)(y)\vert C_{M}'(\vert y\vert + 1)^{-M}$$ and converges to $$T(f)(y)\,\partial _{j}\varphi (y)$$ as h → 0. The Lebesgue dominated convergence theorem yields that the integral on the left-hand side of (2.5.4) converges to

(2.5.5)

Moreover, for a Schwartz function f we have

which converges to ∂ j f(y) pointwise as h → 0 and is bounded by $$C_{M}'(1 + \vert y\vert )^{-M}$$ for $$\vert h\vert < 1/2$$ by an argument similar to the preceding one for $$\varphi$$ in place of f. Thus

(2.5.6)

by the Lebesgue dominated convergence theorem. The boundedness of T from L p to L q yields that

(2.5.7)

Since $$\varphi \in L^{q'}$$, by Hölder’s inequality, the right-hand side of (2.5.4) converges to

as h → 0. This limit is equal to (2.5.5) and the required conclusion follows for $$\alpha = (0,\ldots,0,1,0,\ldots,0)$$. The general case follows by induction on $$\vert \alpha \vert $$. $$\square $$

We now prove Lemma 2.5.4.

Proof.

Let $$R \geq 1$$. Fix a $$\mathcal{C}_{0}^{\infty }$$ function $$\varphi _{R}$$ that is equal to 1 in the ball | x | ≤ R and equal to zero when | x | ≥ 2R. Since h is in $$L^{q}(\mathbf{R}^{n})$$, it follows that $$\varphi _{R}h$$ is in $$L^{1}(\mathbf{R}^{n})$$. We show that $$\widehat{\varphi _{R}h}$$ is also in L ¹. We begin with the inequality

(2.5.8)

which is just a restatement of (2.2.3). Now multiply (2.5.8) by $$\vert \widehat{\varphi _{R}h}(x)\vert $$ to obtain

where we used Leibniz’s rule (Proposition 2.3.22 (14)) and the fact that all derivatives of $$\varphi _{R}$$ are pointwise bounded by constants depending on R.

Integrate the previously displayed inequality with respect to x to obtain

(2.5.9)

Therefore, Fourier inversion holds for $$\varphi _{R}h$$ (see Exercise 2.2.6). This implies that $$\varphi _{R}h$$ is equal a.e. to a continuous function, namely the inverse Fourier transform of its Fourier transform. Since $$\varphi _{R} = 1$$ on the ball B(0, R), we conclude that h is a.e. equal to a continuous function in this ball. Since R > 0 was arbitrary, it follows that h is a.e. equal to a continuous function on $$\mathbf{R}^{n}$$, which we denote by H. Finally, (2.5.2) is a direct consequence of (2.5.9) with R = 1, since $$\vert H(0)\vert \leq \|\widehat{\varphi _{1}h}\|_{L^{1}}$$. □ 

2.5.2 The Transpose and the Adjoint of a Linear Operator

We briefly discuss the notions of the transpose and the adjoint of a linear operator. We first recall real and complex inner products. For f, g measurable functions on $$\mathbf{R}^{n}$$, we define the complex inner product

whenever the integral converges absolutely. We reserve the notation

for the real inner product on $$L^{2}(\mathbf{R}^{n})$$ and also for the action of a distribution f on a test function g. (This notation also makes sense when a distribution f coincides with a function.)

Let $$1 \leq p,q \leq \infty $$. For a bounded linear operator T from $$L^{p}(X,\mu )$$ to L q (Y, ν) we denote by T ∗ its adjoint operator defined by

(2.5.10)

for f in L p (X, μ) and g in L q′(Y, ν) (or in a dense subspace of it). We also define the transpose of T as the unique operator T t that satisfies

for all f ∈ L p (X, μ) and all g ∈ L q′(Y, ν).

If T is an integral operator of the form

then T ∗ and T t are also integral operators with kernels $$K^{{\ast}}(x,y) = \overline{K(y,x)}$$ and K t (x, y) = K(y, x), respectively. If T has the form $$T(f) = (\widehat{f}m)\,^{\vee }\,\,$$, that is, it is given by multiplication on the Fourier transform by a (complex-valued) function $$m(\xi )$$, then T ∗ is given by multiplication on the Fourier transform by the function $$\overline{m(\xi )}$$. Indeed for f, g in $$\mathcal{S}(\mathbf{R}^{n})$$ we have

A similar argument (using Theorem 2.2.14 (5)) gives that if T is given by multiplication on the Fourier transform by the function m(ξ), then T t is given by multiplication on the Fourier transform by the function m(−ξ). Since the complex-valued functions $$\overline{m(\xi )}$$ and m(−ξ) may be different, the operators T ∗ and T t may be different in general. Also, if m(ξ) is real-valued, then T is self-adjoint (i.e., T = T ∗) while if m(ξ) is even, then T is self-transpose (i.e., T = T t ).

2.5.3 The Spaces $$\mathcal{M}^{p,q}(\mathbf{R}^{n})$$

Definition 2.5.5.

Given $$1 \leq p,q \leq \infty $$, we denote by $$\mathcal{M}^{p,q}(\mathbf{R}^{n})$$ the set of all bounded linear operators from $$L^{p}(\mathbf{R}^{n})$$ to $$L^{q}(\mathbf{R}^{n})$$ that commute with translations.

By Theorem 2.5.2 we have that every T in $$\mathcal{M}^{p,q}$$ is given by convolution with a tempered distribution. We introduce a norm on $$\mathcal{M}^{p,q}$$ by setting

that is, the norm of T in $$\mathcal{M}^{p,q}$$ is the operator norm of T as an operator from $$L^{p}$$ to $$L^{q}$$. It is a known fact that under this norm, $$\mathcal{M}^{p,q}$$ is a complete normed space (i.e., a Banach space).

Next we show that when p > q the set $$\mathcal{M}^{p,q}$$ consists of only one element, namely the zero operator T = 0. This means that the only interesting classes of operators arise when p ≤ q.

Theorem 2.5.6.

$$\mathcal{M}^{p,q} =\{ 0\}$$ whenever $$1 \leq q < p < \infty $$ .

Proof.

Let f be a nonzero $$\mathcal{C}_{0}^{\infty }$$ function and let $$h \in \mathbf{R}^{n}$$. We have

Now let $$\vert h\vert \rightarrow \infty $$ and use Exercise 2.5.1. We conclude that

which is impossible if q < p unless T is the zero operator. □ 

Next we have a theorem concerning the duals of the spaces $$\mathcal{M}^{p,q}(\mathbf{R}^{n})$$.

Theorem 2.5.7.

Let $$1 < p \leq q < \infty $$ and $$T \in \mathcal{M}^{p,q}(\mathbf{R}^{n})$$ . Then T can be defined on $$L^{q'}(\mathbf{R}^{n})$$ , coinciding with its previous definition on the subspace $$L^{p}(\mathbf{R}^{n}) \cap L^{q'}(\mathbf{R}^{n})$$ of $$L^{p}(\mathbf{R}^{n})$$ , so that it maps $$L^{q'}(\mathbf{R}^{n})$$ to $$L^{p'}(\mathbf{R}^{n})$$ with norm

(2.5.11)

In other words, we have the following isometric identification of spaces:

Proof.

We first observe that if $$T: L^{p} \rightarrow L^{q}$$ is given by convolution with $$u \in \mathcal{S}'$$, then the adjoint operator $$T^{{\ast}}: L^{q'} \rightarrow L^{p'}$$ is given by convolution with $$\overline{\widetilde{u}} \in \mathcal{S}'$$. Indeed, for $$f,g \in \mathcal{S}(\mathbf{R}^{n})$$ we have

Therefore $$T^{{\ast}}$$ is given by convolution with $$\overline{\widetilde{u}}$$ when applied to Schwartz functions.

Next we observe the validity of the identity

(2.5.12)

It remains to show that T (convolution with u) and T ∗ (convolution with $$\overline{\widetilde{u}}\,$$) map L q′ to L p′ with the same norm. But this easily follows from (2.5.12), which implies that

for all nonzero Schwartz functions f. We conclude that

and therefore

This establishes the claimed assertion. □ 

We next focus attention on the spaces $$\mathcal{M}^{p,q}(\mathbf{R}^{n})$$ whenever p = q. These spaces are of particular interest, since they include the singular integral operators, which we study in Chapter 5.​

2.5.4 Characterizations of $$\mathcal{M}^{1,1}(\mathbf{R}^{n})$$ and $$\mathcal{M}^{2,2}(\mathbf{R}^{n})$$

It would be desirable to have a characterization of the spaces $$\mathcal{M}^{p,p}$$ in terms of properties of the convolving distribution. Unfortunately, this is unknown at present (it is not clear whether it is possible) except for certain cases.

Theorem 2.5.8.

An operator T is in $$\mathcal{M}^{1,1}(\mathbf{R}^{n})$$ if and only if it is given by convolution with a finite Borel (complex-valued) measure. In this case, the norm of the operator is equal to the total variation of the measure.

Proof.

If T is given with convolution with a finite Borel measure μ, then clearly T maps L ¹ to itself and $$\|T\|_{L^{1}\rightarrow L^{1}} \leq \|\mu \|_{\mathcal{M}}$$, where $$\|\mu \|_{\mathcal{M}}$$ is the total variation of μ.

Conversely, let T be an operator bounded from L ¹ to L ¹ that commutes with translations. By Theorem 2.5.2, T is given by convolution with a tempered distribution u. Let

Since the functions $$f_{\varepsilon }$$ are uniformly bounded in L ¹, it follows from the boundedness of $$T$$ that $$f_{\varepsilon } {\ast} u$$ are also uniformly bounded in L ¹. Since L ¹ is naturally embedded in the space of finite Borel measures, which is the dual of the space $$\mathcal{C}_{00}$$ of continuous functions that tend to zero at infinity, we obtain that the family $$f_{\varepsilon } {\ast} u$$ lies in a fixed multiple of the unit ball of $$\mathcal{C}_{00}^{{\ast}}$$. By the Banach–Alaoglu theorem, this is a weak∗ compact set. Therefore, some subsequence of $$f_{\varepsilon } {\ast} u$$ converges in the weak∗ topology to a measure μ. That is, for some $$\varepsilon _{k} \rightarrow 0$$ and all $$g \in \mathcal{C}_{00}(\mathbf{R}^{n})$$ we have

(2.5.13)

We claim that u = μ. To see this, fix $$g \in \mathcal{S}$$. Equation (2.5.13) implies that

as $$k \rightarrow \infty $$. Exercise 2.3.2 gives that $$g {\ast} f_{\varepsilon _{k}}$$ converges to g in $$\mathcal{S}$$. Therefore,

It follows from (2.5.13) that $$\langle u,g\rangle =\langle \mu,g\rangle$$, and since g was arbitrary, u = μ.

Next, (2.5.13) implies that for all $$g \in \mathcal{C}_{00}$$ we have

(2.5.14)

The Riesz representation theorem gives that the norm of the functional

on $$\mathcal{C}_{00}$$ is exactly $$\|\mu \|_{\mathcal{M}}$$. It follows from (2.5.14) that $$\|T\|_{L^{1}\rightarrow L^{1}} \geq \|\mu \|_{\mathcal{M}}$$. Since the reverse inequality is obvious, we conclude that $$\|T\|_{L^{1}\rightarrow L^{1}} =\|\mu \| _{\mathcal{M}}\,.$$ □ 

Let μ be a finite Borel measure. The operator $$h\mapsto h{\ast}\mu$$ maps $$L^{p}(\mathbf{R}^{n})$$ to itself for all $$1 \leq p \leq \infty $$; hence $$\mathcal{M}^{1,1}(\mathbf{R}^{n})$$ can be identified with a subspace of $$\mathcal{M}^{\infty,\infty }(\mathbf{R}^{n})$$. But there exist bounded linear operators Φ on $$L^{\infty }$$ that commute with translations for which there does not exist a finite Borel measure μ such that Φ(h) = h ∗μ for all $$h \in L^{\infty }(\mathbf{R}^{n})$$. The following example captures such a behavior.

Example 2.5.9.

Let $$(X,\|\,\,\cdot \,\,\|_{L^{\infty }})$$ be the space of all complex-valued bounded functions on the real line such that

exists. Then Φ is a bounded linear functional on X with norm 1 and has a bounded extension $$\widetilde{\varPhi }$$ on $$L^{\infty }$$ with norm 1, by the Hahn–Banach theorem. We may view $$\widetilde{\varPhi }$$ as a bounded linear operator from $$L^{\infty }(\mathbf{R})$$ to the space of constant functions, which is contained in $$L^{\infty }(\mathbf{R})$$. We note that $$\widetilde{\varPhi }$$ commutes with translations, since for all $$f \in L^{\infty }(\mathbf{R})$$ and $$x \in \mathbf{R}$$ we have

where the last two equalities follow from the fact that for L ∞ functions f the expression $$\frac{1} {R}\int _{0}^{R}(f(t - x) - f(t))\,dt$$ is bounded by $$\frac{\vert x\vert } {R} \|f\|_{L^{\infty }}$$ when R >  | x | and thus tends to zero as $$R \rightarrow \infty $$. If $$\varPhi (\varphi ) =\varphi {\ast}u$$ for some $$u \in \mathcal{S}'(\mathbf{R}^{n})$$ and all $$\varphi \in \mathcal{S}(\mathbf{R}^{n})$$, since Φ vanishes on $$\mathcal{S}$$, the uniqueness in Theorem 2.5.2 yields that u = 0. Hence, if there existed a finite Borel measure μ such that $$\widetilde{\varPhi }(h) = h{\ast}\mu$$ all $$h \in L^{\infty }$$, in particular we would have $$0 =\varPhi (\varphi ) =\varphi {\ast}\mu$$ for all $$\varphi \in \mathcal{S}$$, hence $$\mu$$ would be the zero measure. But obviously, this is not the case, since $$\varPhi$$ is not the zero operator on X.

We now study the case p = 2. We have the following theorem:

Theorem 2.5.10.

An operator T is in $$\mathcal{M}^{2,2}(\mathbf{R}^{n})$$ if and only if it is given by convolution with some $$u \in \mathcal{S}'$$ whose Fourier transform $$\widehat{u}$$ is an $$L^{\infty }$$ function. In this case the norm of $$T:\, L^{2} \rightarrow L^{2}$$ is equal to $$\|\widehat{u}\,\|_{L^{\infty }}$$ .

Proof.

If $$\widehat{u} \in L^{\infty }$$, Plancherel’s theorem gives

therefore, $$\|T\|_{L^{2}\rightarrow L^{2}} \leq \|\widehat{ u}\,\|_{L^{\infty }}$$, and hence T is in $$\mathcal{M}^{2,2}(\mathbf{R}^{n})$$.

Now suppose that $$T \in \mathcal{M}^{2,2}(\mathbf{R}^{n})$$ is given by convolution with a tempered distribution u. We show that $$\widehat{u}$$ is a bounded function. For $$R > 0$$ let $$\varphi _{R}$$ be a $$\mathcal{C}_{0}^{\infty }$$ function supported inside the ball $$B(0,2R)$$ and equal to one on the ball B(0, R). The product of the function $$\varphi _{R}$$ with the distribution $$\widehat{u}$$ is $$\varphi _{R}\widehat{u} = ((\varphi _{R})\,^{\vee }\,{\ast} u)\,\,\widehat{\,} = T(\varphi _{R}\,^{\vee }\,)\,\,\widehat{\,}\,\,$$, which is an L ² function. Since the L ² function $$\varphi _{R}\widehat{u}$$ coincides with the distribution $$\widehat{u}$$ on the set B(0, R), it follows that $$\widehat{u}$$ is in $$L^{2}(B(0,R))$$ for all R > 0 and therefore it is in $$L_{\mbox{ loc}}^{2}$$. If $$f \in L^{\infty }(\mathbf{R}^{n})$$ has compact support, the function $$f\widehat{u}$$ is in L ², and therefore Plancherel’s theorem and the boundedness of T give

We conclude that for all bounded functions with compact support f we have

Taking $$f(x_{1},\ldots,x_{n}) = (2r)^{-n/2}\prod _{j=1}^{n}\chi _{[-r,r]}(x_{j})$$ for r > 0 and using Corollary 2.1.16, we obtain that $$\|T\|_{L^{2}\rightarrow L^{2}}^{2} -\vert \widehat{u}(x)\vert ^{2} \geq 0$$ for almost all x. Hence $$\widehat{u}$$ is in L ∞ and $$\|\widehat{u}\,\|_{L^{\infty }} \leq \| T\|_{L^{2}\rightarrow L^{2}}$$. Combining this with the estimate $$\|T\|_{L^{2}\rightarrow L^{2}} \leq \|\widehat{ u}\,\|_{L^{\infty }}$$, which holds if $$\widehat{u} \in L^{\infty }$$, we deduce that $$\|T\|_{L^{2}\rightarrow L^{2}} =\|\widehat{ u}\,\|_{L^{\infty }}$$. □ 

2.5.5 The Space of Fourier Multipliers $$\mathcal{M}_{p}(\mathbf{R}^{n})$$

We have now characterized all convolution operators that map L ² to L ². Suppose now that T is in $$\mathcal{M}^{p,p}$$, where 1 < p < 2. As discussed in Theorem 2.5.7, T also maps L p′ to L p′. Since p < 2 < p′, by Theorem 1.​3.​4, it follows that T also maps L ² to L ². Thus T is given by convolution with a tempered distribution whose Fourier transform is a bounded function.

Definition 2.5.11.

Given $$1 \leq p < \infty $$, we denote by $$\mathcal{M}_{p}(\mathbf{R}^{n})$$ the space of all bounded functions m on $$\mathbf{R}^{n}$$ such that the operator

is bounded on $$L^{p}(\mathbf{R}^{n})$$ (or is initially defined in a dense subspace of $$L^{p}(\mathbf{R}^{n})$$ and has a bounded extension on the whole space). The norm of m in $$\mathcal{M}_{p}(\mathbf{R}^{n})$$ is defined by

(2.5.15)

Definition 2.5.11 implies that $$m \in \mathcal{M}_{p}$$ if and only if $$T_{m} \in \mathcal{M}^{p,p}$$. Elements of the space $$\mathcal{M}_{p}$$ are called L p multipliers or L p Fourier multipliers. It follows from Theorem 2.5.10 that $$\mathcal{M}_{2}$$, the set of all L ² multipliers, is $$L^{\infty }$$. Theorem 2.5.8 implies that $$\mathcal{M}_{1}(\mathbf{R}^{n})$$ is the set of the Fourier transforms of finite Borel measures that is usually denoted by $$\mathcal{M}(\mathbf{R}^{n})$$. Theorem 2.5.7 states that a bounded function m is an L p multiplier if and only if it is an L p′ multiplier, and in this case

It is a consequence of Theorem 1.​3.​4 that the normed spaces $$\mathcal{M}_{p}$$ are nested, that is, for $$1 \leq p \leq q \leq 2$$ we have

Moreover, if $$m \in \mathcal{M}_{p}$$ and $$1 \leq p \leq 2 \leq p'$$, Theorem 1.​3.​4 gives

(2.5.16)

since $$1/2 = (1/2)/p + (1/2)/p'$$. Theorem 1.​3.​4 also gives that

whenever $$1 \leq q \leq p \leq 2$$. Thus the $$\mathcal{M}_{p}$$’s form an increasing family of spaces as p increases from 1 to 2.

Example 2.5.12.

The function $$m(\xi ) = e^{2\pi i\xi \cdot b}$$ is an L p multiplier for all $$b \in \mathbf{R}^{n}$$, since the corresponding operator $$T_{m}(f)(x) = f(x + b)$$ is bounded on $$L^{p}(\mathbf{R}^{n})$$. Clearly $$\|m\|_{\mathcal{M}_{p}} = 1$$.

Proposition 2.5.13.

For $$1 \leq p < \infty $$ , the normed space $$\big(\mathcal{M}_{p},\|\,\cdot \,\|_{\mathcal{M}_{p}}\big)$$ is a Banach space. Furthermore, $$\mathcal{M}_{p}$$ is closed under pointwise multiplication and is a Banach algebra.

Proof.

It suffices to consider the case $$1 \leq p \leq 2$$. It is straightforward that if m ₁, m ₂ are in $$\mathcal{M}_{p}$$ and $$b \in \mathbf{C}$$ then $$m_{1} + m_{2}$$ and bm ₁ are also in $$\mathcal{M}_{p}$$. Observe that $$m_{1}m_{2}$$ is the multiplier that corresponds to the operator $$T_{m_{1}}T_{m_{2}} = T_{m_{1}m_{2}}$$ and thus

This proves that $$\mathcal{M}_{p}$$ is an algebra. To show that $$\mathcal{M}_{p}$$ is a complete normed space, consider a Cauchy sequence $$m_{j}$$ in $$\mathcal{M}_{p}$$. It follows from (2.5.16) that m j is Cauchy in $$L^{\infty }$$, and hence it converges to some bounded function m in the $$L^{\infty }$$ norm; moreover all the m j are a.e. bounded by some constant C uniformly in j. We have to show that $$m \in \mathcal{M}_{p}$$. Fix $$f \in \mathcal{S}$$. We have

a.e. by the Lebesgue dominated convergence theorem, since $$C\,\vert \widehat{f}\,\vert $$ is an integrable upper bound of all integrands on the left in the preceding expression. Since $$\{m_{j}\}_{j}$$ is a Cauchy sequence in $$\mathcal{M}_{p}$$, it is bounded in $$\mathcal{M}_{p}$$, and thus $$\sup _{j}\|m_{j}\|_{\mathcal{M}_{p}} < +\infty.$$ An application of Fatou’s lemma yields that

which implies that $$m \in \mathcal{M}_{p}$$. This argument shows that if $$m_{j} \in \mathcal{M}_{p}$$ and $$m_{j} \rightarrow m$$ uniformly, then m is in $$\mathcal{M}_{p}$$ and satisfies

Apply this inequality to $$m_{k} - m_{j}$$ in place of m j and m k − m in place of m, for some fixed k. We obtain

(2.5.17)

for each k. Given $$\varepsilon > 0$$, by the Cauchy criterion, there is an N such that for j, k > N we have $$\|m_{k} - m_{j}\|_{\mathcal{M}_{p}} <\varepsilon$$. Using (2.5.17) we conclude that $$\|m_{k} - m\|_{\mathcal{M}_{p}} \leq \varepsilon$$ when k > N, thus m k converges to m in $$\mathcal{M}_{p}$$.

This proves that $$\mathcal{M}_{p}$$ is a Banach space. □ 

The following proposition summarizes some simple properties of multipliers.

Proposition 2.5.14.

For all $$m \in \mathcal{M}_{p}$$ , $$1 \leq p < \infty $$ , $$x \in \mathbf{R}^{n}$$ , and h > 0 we have

(2.5.18)

(2.5.19)

Proof.

See Exercise 2.5.2. □ 

Example 2.5.15.

We show that for $$-\infty \! <\! a\! <\! b\! <\! \infty $$ we have $$\|\chi _{[a,b]}\|_{\mathcal{M}_{p}} =\|\chi _{[0,1]}\|_{\mathcal{M}_{p}}$$. Indeed, using (2.5.18) we obtain that $$\|\chi _{[a,b]}\|_{\mathcal{M}_{p}} =\|\chi _{[0,b-a]}\|_{\mathcal{M}_{p}}\,,$$ and the latter is equal to $$\|\chi _{[0,1]}\|_{\mathcal{M}_{p}}$$ in view of (2.5.19). The fact that we have $$\|\chi _{[0,1]}\|_{\mathcal{M}_{p}} < \infty $$ for all $$1 < p < \infty $$ is shown in Chapter 5.​

We continue with the following interesting result.

Theorem 2.5.16.

Suppose that $$m(\xi,\eta ) \in \mathcal{M}_{p}(\mathbf{R}^{n+m})$$ , where $$1 < p < \infty $$ . Then for almost every $$\xi \in \mathbf{R}^{n}$$ the function $$\eta \mapsto m(\xi,\eta )$$ is in $$\mathcal{M}_{p}(\mathbf{R}^{m})$$ , with

Proof.

Since m lies in $$L^{\infty }(\mathbf{R}^{n+m})$$, it follows by Fubini’s theorem that for almost all $$\xi \in \mathbf{R}^{n}$$, the function $$\eta \mapsto m(\xi,\eta )$$ lies in $$L^{\infty }(\mathbf{R}^{m})$$ and

(2.5.20)

Fix f ₁, g ₁ in $$\mathcal{S}(\mathbf{R}^{n})$$ and f ₂, g ₂ in $$\mathcal{S}(\mathbf{R}^{m})$$. Define the functions $$(f_{1} \otimes f_{2})(x,y) = f_{1}(x)f_{2}(y)$$ when $$x \in \mathbf{R}^{n}$$ and $$y \in \mathbf{R}^{m}$$. For all $$\xi$$ for which (2.5.20) is satisfied define

and observe that

In view of the identity

it follows that, for the $$\xi$$ that satisfy (2.5.20), M(ξ) lies in $$\mathcal{M}_{p}(\mathbf{R}^{n})$$ with

Since $$\|M\|_{L^{\infty }} \leq \| M\|_{\mathcal{M}_{p}}$$ for almost all $$\xi \in \mathbf{R}^{n}$$, we obtain

(2.5.21)

which of course implies the required conclusion, by taking the supremum over all g ₂ in L p′ with norm at most 1. □ 

Example 2.5.17.

(The cone multiplier) On R n+1 define the function

where the plus sign indicates that $$m_{\lambda } = 0$$ if the expression inside the parentheses is negative. The multiplier $$m_{\lambda }$$ is called the cone multiplier with parameter $$\lambda$$. If $$m_{\lambda }$$ is in $$\mathcal{M}_{p}(\mathbf{R}^{n+1})$$, then the function $$b_{\lambda }(\xi ) = (1 -\vert \xi \vert ^{2})_{+}^{\lambda }$$ defined on $$\mathbf{R}^{n}$$ is in $$\mathcal{M}_{p}(\mathbf{R}^{n})$$. Indeed, by Theorem 2.5.16 we have that for some $$\xi _{n+1} = h$$, $$b_{\lambda }(\xi _{1}/h,\ldots,\xi _{n}/h)$$ is in $$\mathcal{M}_{p}(\mathbf{R}^{n})$$ and hence so is $$b_{\lambda }$$ by property (2.5.19).

Exercises

2.5.1. Prove that if $$f \in L^{q}(\mathbf{R}^{n})$$ and $$0 < q < \infty $$, then

2.5.2. Prove Proposition 2.5.14. Also prove that if $$\delta _{j}^{h_{j}}$$ is a dilation operator in the jth variable (for instance $$\delta _{1}^{h_{1}}f(x) = f(h_{1}x_{1},x_{2},\ldots,x_{n})$$), then

2.5.3. Let $$m \in \mathcal{M}_{p}(\mathbf{R}^{n})$$ where $$1 \leq p < \infty $$. (a) If ψ is a function on $$\mathbf{R}^{n}$$ whose inverse Fourier transform is an integrable function, then prove that

(b) If $$\psi$$ is in $$L^{1}(\mathbf{R}^{n})$$, then prove that

2.5.4. Fix a multi-index $$\gamma$$.

(a) Prove that the map $$T(f) = f {\ast} \partial ^{\gamma }\delta _{0}$$ maps $$\mathcal{S}$$ continuously into $$\mathcal{S}$$.

(b) Prove that when $$1/p - 1/q\neq \vert \gamma \vert /n$$, T does not extend to an element of the space $$\mathcal{M}^{p,q}$$.

2.5.5.

Let $$K_{\gamma }(x) = \vert x\vert ^{-n+\gamma }$$, where $$0 <\gamma < n$$. Use Theorem 1.​4.​25 to show that the operator

extends to a bounded operator in $$\mathcal{M}^{p,q}(\mathbf{R}^{n})$$, where $$1/p - 1/q =\gamma /n$$, $$1 < p < q < \infty $$. This provides an example of a nontrivial operator in $$\mathcal{M}^{p,q}(\mathbf{R}^{n})$$ when p < q.

2.5.6.

(a) Use the ideas of the proof of Proposition 2.5.13 to show that if $$m_{j} \in \mathcal{M}_{p}$$, $$1 \leq p < \infty $$, $$\|m_{j}\|_{\mathcal{M}_{p}} \leq C$$ for all $$j = 1,2,\ldots$$, and m j  → m a.e., then $$m \in \mathcal{M}_{p}$$ and

(b) Prove that if $$m \in \mathcal{M}_{p}$$, $$1 \leq p < \infty $$, and the limit $$m_{0}(\xi ) =\lim \limits _{R\rightarrow \infty }m(\xi /R)$$ exists for all $$\xi \in \mathbf{R}^{n}$$, then $$m_{0}$$ is a radial function in $$\mathcal{M}_{p}(\mathbf{R}^{n})$$ and satisfies $$\|m_{0}\|_{\mathcal{M}_{p}} \leq \| m\|_{\mathcal{M}_{p}}$$.

(c) If $$m \in \mathcal{M}_{p}(\mathbf{R})$$ has left and right limits at the origin, then prove that

(d) Suppose that for some $$1 \leq p < \infty $$, $$m_{t} \in \mathcal{M}_{p}(\mathbf{R}^{n})$$ for all $$0 < t < \infty $$. Prove that

2.5.7.

Let 1 ≤ p < ∞ and suppose that $$m \in \mathcal{M}_{p}(\mathbf{R}^{n})$$ satisfies $$\vert m(\xi )\vert \geq c\,(1 + \vert \xi \vert )^{-N}$$ for some c, N > 0. Prove that the operator $$T(f) = (\widehat{f}\,m^{-1})\,^{\vee }\,\,$$ satisfies $$\|T(f)\|_{L^{p}} \geq c_{p}\|f\|_{L^{p}}$$ for all $$f \in \mathcal{S}(\mathbf{R}^{n})$$, where $$c_{p} =\| m\|_{\mathcal{M}_{p}}^{-1}$$.

2.5.8.

(a) Prove that if $$m \in L^{\infty }(\mathbf{R}^{n})$$ satisfies $$m\,^{\vee }\,\geq 0$$, then for all $$1 \leq p < \infty $$ we have

(b) (L. Colzani and E. Laeng) On the real line let

Prove that

for all $$1 < p < \infty $$.

[Hint: Part (a): Use Exercise 1.2.9. Part (b): Use part (a) to show that $$\|m_{2}m_{1}^{-1}\|_{\mathcal{M}_{p}} = 1$$. Deduce that $$\|m_{2}\|_{\mathcal{M}_{p}} \leq \| m_{1}\|_{\mathcal{M}_{p}}$$. For the converse use Exercise 2.5.6 (c).]

2.5.9. ([94]) Let $$1 < p < \infty $$ and $$0 < A < \infty $$. Prove that the following are equivalent:

(a) The operator $$f\mapsto \sum _{m\in \mathbf{Z}^{n}}a_{m}f(x - m)$$ is bounded on $$L^{p}(\mathbf{R}^{n})$$ with norm A.

(b) The $$\mathcal{M}_{p}$$ norm of the function $$\sum _{m\in \mathbf{Z}^{n}}a_{m}e^{-2\pi im\cdot x}$$ is exactly A.

(c) The operator given by convolution with the sequence {a m } is bounded on $$\ell^{p}(\mathbf{Z}^{n})$$ with norm A.

2.5.10. ([177]) Let $$m(\xi )$$ in $$\mathcal{M}_{p}(\mathbf{R}^{n})$$ be supported in [0, 1] n . Then the periodic extension of m in $$\mathbf{R}^{n}$$,

is also in $$\mathcal{M}_{p}(\mathbf{R}^{n})$$.

2.5.11. Suppose that u is a $$\mathcal{C}^{\infty }$$ function on $$\mathbf{R}^{n}\setminus \{0\}$$ that is homogeneous of degree $$-n + i\tau$$, $$\tau \in \mathbf{R}$$. Prove that the operator given by convolution with u maps $$L^{2}(\mathbf{R}^{n})$$ to $$L^{2}(\mathbf{R}^{n})$$.

2.5.12. ([142]) Let $$m_{1} \in L^{r}(\mathbf{R}^{n})$$ and $$m_{2} \in L^{r'}(\mathbf{R}^{n})$$ for some $$2 \leq r \leq \infty $$. Prove that $$m_{1} {\ast} m_{2} \in \mathcal{M}_{p}(\mathbf{R}^{n})$$ when $$\frac{1} {p} -\frac{1} {2} = \frac{1} {r}$$ and $$1 \leq p \leq 2$$.

[Hint: Prove that the trilinear operator $$(m_{1},m_{2},f)\!\mapsto \!\big((m_{1} {\ast} m_{2})\widehat{f}\,\,\big)\!\,^{\vee }\,$$ is bounded from $$L^{2} \times L^{2} \times L^{1} \rightarrow L^{1}$$ and $$L^{\infty }\times L^{1} \times L^{2} \rightarrow L^{2}$$. Apply trilinear complex interpolation (Corollary 7.2.11 in [131]) to deduce the required conclusion for 1 ≤ p ≤ 2.]

2.5.13. Show that the function $$e^{i\vert \xi \vert ^{2} }$$ is an L p Fourier multiplier on $$\mathbf{R}^{n}$$ if and only if p = 2.

[Hint: By Exercise 2.4.12 the inverse Fourier transform of $$e^{i\vert \xi \vert ^{2} }$$ is in $$L^{\infty }$$, thus the operator $$f\mapsto \big(\widehat{f}(\xi )e^{i\pi \vert \xi \vert ^{2} }\big)\,^{\vee }\,$$ maps L ¹ to $$L^{\infty }$$. Since this operator also maps L ² to L ², it should map L p to L p′ for all 1 ≤ p ≤ 2.]

2.6 Oscillatory Integrals

Oscillatory integrals have played an important role in harmonic analysis from its outset. The Fourier transform is the prototype of oscillatory integrals and provides the simplest example of a nontrivial phase, a linear function of the variable of integration. More complicated phases naturally appear in the subject; for instance, Bessel functions provide examples of oscillatory integrals in which the phase is a sinusoidal function.

In this section we take a quick look at oscillatory integrals. We mostly concentrate on one-dimensional results, which already require some significant analysis. We examine only a very simple higher-dimensional situation. Our analysis here is far from adequate.

Definition 2.6.1.

An oscillatory integral is an expression of the form

(2.6.1)

where $$\lambda$$ is a positive real number, $$\varphi$$ is a real-valued function on $$\mathbf{R}^{n}$$ called the phase , and $$\psi$$ is a complex-valued and smooth integrable function on $$\mathbf{R}^{n}$$, which is often taken to have compact support.

2.6.1 Phases with No Critical Points

We begin by studying the simplest possible one-dimensional case. Suppose that $$\varphi$$ and ψ are smooth functions on the real line such that supp ​ψ is a closed interval and

Since $$\varphi '$$ has no zeros, it must be either strictly positive or strictly negative everywhere on the support of ψ. It follows that $$\varphi$$ is monotonic on the support of ψ and we are allowed to change variables

in (2.6.1). Then $$dx = (\varphi '(x))^{-1}du = (\varphi ^{-1})'(u)\,du$$, where $$\varphi ^{-1}$$ is the inverse function of $$\varphi$$. We transform the integral in (2.6.1) into

(2.6.2)

and we note that the function $$\theta (u) =\psi (\varphi ^{-1}(u))(\varphi ^{-1})'(u)$$ is smooth and has compact support on $$\mathbf{R}$$. We therefore interpret the integral in (2.6.1) as $$\widehat{\theta }(-\lambda /2\pi )$$, where $$\widehat{\theta }$$ is the Fourier transform of θ. Since θ is a smooth function with compact support, it follows that the integral in (2.6.2) has rapid decay as $$\lambda \rightarrow \infty $$.

A quick way to see that the expression $$\widehat{\theta }(-\lambda /2\pi )$$ has decay of order $$\lambda ^{-N}$$ for all N > 0 as $$\lambda$$ tends to $$\infty $$ is the following. Write

and integrate by parts N times to express the integral in (2.6.2) as

from which the assertion follows. Hence

(2.6.3)

where $$C_{N} =\|\theta ^{(N)}\|_{L^{1}}$$, which depends on derivatives of $$\varphi$$ and ψ.

We now turn to a higher-dimensional analogue of this situation.

Definition 2.6.2.

We say that a point x ₀ is a critical point of a phase function $$\varphi$$ if

Example 2.6.3.

Let $$\xi \in \mathbf{R}^{n}\setminus \{0\}$$. Then the phase functions $$\varphi _{1}(x) = x\cdot \xi$$, $$\varphi _{2}(x) = e^{x\cdot \xi }$$ have no critical points, while the phase function $$\varphi _{3}(x) = \vert x\vert ^{2} - x\cdot \xi$$ has one critical point at $$x_{0} = \frac{1} {2}\xi$$.

The next result concerns the behavior of oscillatory integrals whose phase functions have no critical points.

Proposition 2.6.4.

Suppose that ψ is a compactly supported smooth function on $$\mathbf{R}^{n}$$ and that $$\varphi$$ is a real-valued $$\mathcal{C}^{\infty }$$ function on $$\mathbf{R}^{n}$$ that has no critical points on the support of ψ. Then the oscillatory integral

(2.6.4)

obeys a bound of the form $$\vert I(\lambda )\vert \leq C_{N}\lambda ^{-N}$$ for all $$\lambda \geq 1$$ and all N > 0, where C N depends on N and on $$\varphi$$ and ψ.

Proof.

Since the case n = 1 has already been discussed, we concentrate on dimensions n ≥ 2. For each y in the support of ψ there is a unit vector θ y such that

By the continuity of $$\nabla \varphi$$ there is a small neighborhood $$B(y,r_{y})$$ of y such that for all $$x \in B(y,r_{y})$$ we have

Cover the support of ψ by a finite number of balls $$B(y_{j},r_{y_{j}})$$, $$j = 1,\ldots,m$$, and pick $$c =\min _{j}\frac{1} {2}\vert \nabla \varphi (y_{j})\vert $$; we have

(2.6.5)

for all $$x \in B(y_{j},r_{y_{j}})$$ and $$j = 1,\ldots,m$$.

Next we find a smooth partition of unity of $$\mathbf{R}^{n}$$ such that each member ζ k of the partition is supported in some ball $$B(y_{j},r_{y_{j}})$$ or lies outside the support of ψ. We therefore write

(2.6.6)

where the sum contains only a finite number of indices, since only a finite number of the ζ k ’s meet the support of ψ. It suffices to show that every term in the sum in (2.6.6) has rapid decay in $$\lambda$$ as $$\lambda \rightarrow \infty $$.

To this end, we fix a k and we pick a j such that the support of $$\psi \zeta _{k}$$ is contained in some ball $$B(y_{j},r_{y_{j}})$$. We find unit vectors $$\theta _{y_{j},2},\ldots,\theta _{y_{j},n}$$, such that the system $$\{\theta _{y_{j}},\theta _{y_{j},2},\ldots,\theta _{y_{j},n}\}$$ is an orthonormal basis of $$\mathbf{R}^{n}$$. Let e j be the unit (column) vector on R n whose jth coordinate is one and whose remaining coordinates are zero. We find an orthogonal matrix R such that $$R^{t}e_{1} =\theta _{y_{j}}$$ and we introduce the change of variables $$u = y_{j} + R(x - y_{j})$$ in the integral

The map $$x\mapsto u = (u_{1},\ldots,u_{n})$$ is a rotation that fixes y j and preserves the ball $$B(y_{j},r_{y_{j}})$$. Defining $$\varphi (x) =\varphi ^{o}(u)$$, $$\psi (x) =\psi ^{o}(u)$$, $$\zeta _{k}(x) =\zeta _{ k}^{o}(u)$$, under this new coordinate system we write

(2.6.7)

where K is a compact subset of $$\mathbf{R}^{n-1}$$. Since R is an orthogonal matrix, $$R^{-1} = R^{t}$$, and the change of variables $$x = y_{j} + R^{t}(u - y_{j})$$ implies that

Thus for all $$x \in B(y_{j},r_{j})$$ we have

in view of condition (2.6.5). This lower estimate is valid for all $$u \in B(y_{j},r_{y_{j}})$$, and therefore the inner integral inside the curly brackets in (2.6.7) is at most $$C_{N}\lambda ^{-N}$$ by estimate (2.6.3). Integrating over K results in the same conclusion for $$I(\lambda )$$ defined in (2.6.4). □ 

2.6.2 Sublevel Set Estimates and the Van der Corput Lemma

We discuss a sharp decay estimate for one-dimensional oscillatory integrals. This estimate is obtained as a consequence of delicate size estimates for the Lebesgue measures of the sublevel sets $$\{\vert u\vert \leq \alpha \}$$ for a function u. In what follows, u (k) denotes the kth derivative of a function u(t) defined on $$\mathbf{R}$$, and $$\mathcal{C}^{k}$$ the space of all functions whose kth derivative exists and is continuous.

Lemma 2.6.5.

Let k ≥ 1 and suppose that $$a_{0},\ldots,a_{k}$$ are distinct real numbers. Let a = min (a j ) and b = max (a j ) and let f be a real-valued $$\mathcal{C}^{k-1}$$ function on [a,b] that is $$\mathcal{C}^{k}$$ on (a,b). Then there exists a point y in (a,b) such that

where $$c_{m} = (-1)^{k}\,k!\,\prod \limits _{\begin{array}{c}\ell=0 \\ \ell\neq m \end{array}}^{k}(a_{\ell} - a_{m})^{-1}$$ .

Proof.

Suppose we could find a polynomial $$p_{k}(x) =\sum _{ j=0}^{k}b_{j}x^{j}$$ such that the function

(2.6.8)

satisfies $$\varphi (a_{m}) = 0$$ for all 0 ≤ m ≤ k. Since the a j are distinct, we apply Rolle’s theorem k times to find a point y in (a, b) such that

The existence of a polynomial p k such that (2.6.8) is satisfied is equivalent to the existence of a solution to the matrix equation

The determinant of the square matrix on the left is called the Vandermonde determinant and is equal to

Since the a j are distinct, it follows that the system has a unique solution. Using Cramer’s rule, we solve this system to obtain

The required conclusion now follows with c m as claimed. □ 

Lemma 2.6.6.

Let E be a measurable subset of $$\mathbf{R}$$ with finite nonzero Lebesgue measure and let $$k \in \mathbf{Z}^{+}$$ . Then there exist $$a_{0},\ldots,a_{k}$$ in E such that for all $$\ell= 0,1,\ldots,k$$ we have

(2.6.9)

Proof.

Given a measurable set E with finite measure, pick a compact subset E′ of E such that $$\vert E\setminus E'\vert <\delta$$, for some $$\delta > 0$$. For $$x \in \mathbf{R}$$ define $$T(x) = \vert (-\infty,x) \cap E'\vert $$. Then T enjoys the distance-decreasing property

for all $$x,y \in E'$$; consequently, by the intermediate value theorem, T is a surjective map from E′ to [0, | E′ | ]. Let a j be points in E′ such that $$T(a_{j}) = \frac{j} {k}\vert E'\vert $$ for $$j = 0,1,\ldots,k$$. For k an even integer, we have

and it is easily shown that $$\big((k/2)!\big)^{2}k^{-k} \geq (2e)^{-k}$$.

For k an odd integer we have

while the last product is at least

We have therefore proved (2.6.9) with E′ replacing E. Since $$\vert E\setminus E'\vert <\delta$$ and $$\delta > 0$$ is arbitrarily small, the required conclusion follows. □ 

The following is the main result of this section.

Proposition 2.6.7.

(a) Let u be a real-valued $$\mathcal{C}^{k}$$ function, $$k \in \mathbf{Z}^{+}$$ , that satisfies u (k) (t) ≥ 1 for all $$t \in \mathbf{R}$$ . Then the following estimate is valid for all $$\alpha > 0$$ :

(2.6.10)

(b) Let $$-\infty < a < b < \infty $$ . For all k ≥ 2, for every real-valued $$\mathcal{C}^{k}$$ function u on the line that satisfies $$u^{(k)}(t) \geq 1$$ for all t ∈ [a,b], and every $$\lambda \in \mathbf{R}\setminus \{0\}$$ we have:

(2.6.11)

(c) If k = 1, u′(t) is monotonic on (a,b), and u′(t) ≥ 1 for all t ∈ (a,b), then for all nonzero real numbers $$\lambda$$ we have

(2.6.12)

Proof.

Part (a): Let $$E =\{ t \in \mathbf{R}:\,\, \vert u(t)\vert \leq \alpha \}$$. If | E | is nonzero, then by Lemma 2.6.6 there exist $$a_{0},a_{1},\ldots,a_{k}$$ in E such that for all ℓ we have

(2.6.13)

Lemma 2.6.5 implies that there exists $$y \in \big (\min a_{j},\max a_{j}\big)$$ such that

(2.6.14)

Using (2.6.13), we obtain that the expression on the right in (2.6.14) is in absolute value at most

since a j  ∈ E. The bound u (k)(t) ≥ 1 now implies

as claimed. This proves (2.6.10).

Part (b): We now take k ≥ 2 and we split the interval (a, b) in (2.6.11) into the sets

for some parameter $$\beta$$ to be chosen momentarily. The function v = u′ satisfies $$v^{(k-1)} \geq 1$$ and k − 1 ≥ 1. It follows from part (a) that

To obtain the corresponding estimate over R ₂, we note that if u (k) ≥ 1, then the set $$\{\vert u'\vert >\beta \}$$ is the union of at most 2k − 2 intervals on each of which u′ is monotone. Let (c, d) be one of these intervals on which u′ is monotone. Then u′ has a fixed sign on (c, d) and we have

where we use the monotonicity of 1∕u′(t) in moving the absolute value from inside the integral to outside. It follows that

Choosing $$\beta = \vert \lambda \vert ^{-(k-1)/k}$$ to optimize and adding the corresponding estimates for R ₁ and R ₂, we deduce the claimed estimate (2.6.11).

Part (c): Repeat the argument in part (b) setting $$\beta = 1$$ and replacing the interval (c, d) by (a, b). □ 

Corollary 2.6.8.

Let (a,b), u(t), $$\lambda > 0$$ , and k be as in Proposition 2.6.7. Then for any function ψ on (a,b) with an integrable derivative and k ≥ 2, we have

We also have

when k = 1 and u′ is monotonic on (a,b).

Proof.

Set

and use integration by parts to write

The conclusion easily follows. □ 

Example 2.6.9.

The Bessel function of order m is defined as

Here we take both r and m to be real numbers, and we suppose that $$m > -\frac{1} {2}$$; we refer to Appendix B for an introduction to Bessel functions and their basic properties.

We use Corollary 2.6.8 to calculate the decay of the Bessel function J m (r) as $$r \rightarrow \infty $$. Set

and note that $$\varphi '(\theta )$$ vanishes only at $$\theta =\pi /2$$ and $$3\pi /2$$ inside the interval $$[0,2\pi ]$$ and that $$\varphi ''(\pi /2) = -1$$, while $$\varphi ''(3\pi /2) = 1.$$ We now write $$1 =\psi _{1} +\psi _{2} +\psi _{3}$$, where ψ ₁ is smooth and compactly supported in a small neighborhood of π∕2, and ψ ₂ is smooth and compactly supported in a small neighborhood of 3π∕2. For j = 1, 2, Corollary 2.6.8 yields

for some constant C, while the corresponding integral containing ψ ₃ has arbitrary decay in r in view of estimate (2.6.3) (or Proposition 2.6.4 when n = 1).

Exercises

2.6.1. Suppose that u is a real-valued $$\mathcal{C}^{k}$$ function defined on the line that satisfies $$\vert u^{(k)}(t)\vert \geq c_{0} > 0$$ for some k ≥ 2 and all t ∈ (a, b). Prove that for $$\lambda \in \mathbf{R}\setminus \{0\}$$ we have

and that the same conclusion is valid when k = 1, provided u′ is monotonic.

2.6.2. Show that if u′ is not monotonic in part (c) of Proposition 2.6.7, then the conclusion may fail.

[Hint: Let $$\varphi (t)$$ be a real-valued smooth function that is equal to 2t on intervals $$[2\pi k +\varepsilon _{k},2\pi (k + \frac{1} {2}) -\varepsilon _{k}]$$ and equal to t on intervals $$[2\pi (k + \frac{1} {2}) +\varepsilon _{k},2\pi (k + 1) -\varepsilon _{k}]$$, where $$0 \leq k \leq N$$, for some $$N \in \mathbf{Z}^{+}$$. Show that the absolute value of the integral of $$e^{i\varphi (t)}$$ over the interval $$[\varepsilon _{0},2\pi (N + 1) -\varepsilon _{N}]$$ tends to infinity as $$N \rightarrow \infty $$.]

2.6.3. Prove that the dependence on k of the constant in part (b) of Proposition 2.6.7 is indeed linear.

[Hint: Take u(t) = t k ∕k! over the interval (0, k! ).]

2.6.4. Follow the steps below to give an alternative proof of part (b) of Proposition 2.6.7. Assume that the statement is known for some k ≥ 2 and some constant C(k) for all intervals [a, b] and all $$\mathcal{C}^{k}$$ functions satisfying u (k) ≥ 1 on [a, b]. Fix a $$\mathcal{C}^{k+1}$$ function u such that u (k+1) ≥ 1 on an interval [a, b]. Let c be the unique point at which the function u (k) attains its minimum in [a, b]. (a) If u (k)(c) = 0, then for all $$\delta > 0$$ we have $$u^{(k)}(t) \geq \delta$$ in the complement of the interval $$(c-\delta,c+\delta )$$ and derive the bound

(b) If $$u^{(k)}(c)\neq 0$$, then we must have $$c \in \{ a,b\}$$. Obtain the bound

(c) Choose a suitable $$\delta$$ to optimize and deduce the validity of the statement for k + 1 with C(k + 1) = 2C(k) + 2, hence $$C(k) = 3 \cdot 2^{k-1} + 2^{k} - 2$$, since C(1) = 3.

2.6.5. (a) Prove that for some constant C and all $$\lambda \in \mathbf{R}$$ and $$\varepsilon \in (0,1)$$ we have

(b) Prove that for some , all , , and we have

(c) Show that there is a constant C″ such that for any , for all in , and for all integers k ≥ 2, we have

[Hint: Part (a): For small use the inequality . If is large, split the domains of integration into the regions and and use integration by parts in the second case. Part (b): Write

and use part (a). Part (c): When it is trivial. If ξ ₂ = 0, ξ ₁ ≠ 0, change variables t = ξ ₁ s and then split the domain of integration into the sets | t | ≤ 1 and | t | ≥ 1. In the interval over the set | t | ≤ 1 apply part (b) and over the set | t | ≥ 1 use integration by parts. In the case ξ ₂ ≠ 0, change variables and split the domain of integration into the sets | t | ≥ 1 and | t | ≤ 1. When | t | ≤ 1 use part (b) and in the case | t | ≥ 1 use Corollary 2.6.8, noting that .]

2.6.6. (a) Show that for all a > 0 and the following is valid:

(b) Prove that there is a constant c > 0 such that for all we have

[Hint: Part (b): Consider the intervals (0, δ) and for some . Apply Proposition 2.6.7 with k = 1 on one of these intervals and with k = 2 on the other. Then choose a suitable .]

2.6.7. Show that there is a constant such that for all nonintegers and all we have

[Hint: On the interval apply Proposition 2.6.7 with and on the interval with . Then optimize by choosing .]

HISTORICAL NOTES

The one-dimensional maximal function originated in the work of Hardy and Littlewood [146]. Its n-dimensional analogue was introduced by Wiener [375], who used Lemma 2.1.5, a variant of the Vitali covering lemma, to derive its L p boundedness. One may consult the books of de Guzmán [92], [93] for extensions and other variants of such covering lemmas. The actual covering lemma proved by Vitali [368] says that if a family of closed cubes in has the property that for every point there exists a sequence of cubes in the family that tends to x, then it is always possible to extract a sequence of pairwise disjoint cubes from the family such that . We refer to Saks [310] for details and extensions of this theorem.

The class was introduced by Zygmund to give a sufficient condition on the local integrability of the Hardy–Littlewood maximal operator. The necessity of this condition was observed by Stein [336]. Stein [341] also showed that the norm of the centered Hardy–Littlewood maximal operator is bounded above by some dimension-free constant; see also Stein and Strömberg [345]. Analogous results for maximal operators associated with convex bodies are contained in Bourgain [35], Carbery [51], and Müller [263]. Bourgain [37] showed the the Hardy-Littlewood maximal operator associated with cubes is bounded on with dimension-free bounds when p > 1. Aldaz [2] studied the corresponding weak type (1, 1) bounds and proved that they grow to infinity with the dimension; the constant was improved by Aubrun [15]. The situation for the uncentered maximal operator M on L p is different, since given any there exists C p  > 1 such that (see Exercise 2.1.8 for a value of such a constant C p and also the article of Grafakos and Montgomery-Smith [136] for a larger value).

The centered maximal function with respect to a general inner regular locally finite positive measure μ on is bounded on without the additional hypothesis that the measure is doubling; see Fefferman [117]. The proof of this result requires the following covering lemma, obtained by Besicovitch [27]: Given any family of closed balls whose centers form a bounded subset of , there exists an at most countable subfamily of balls that covers the set of centers and has bounded overlap, i.e., no point in belongs to more than a finite number (depending on the dimension) of the balls in the subfamily. A similar version of this lemma was obtained independently by Morse [258]. See also Ziemer [385] for an alternative formulation. The uncentered maximal operator M μ of Exercise 2.1.1 may not be weak type (1, 1) if the measure μ is nondoubling, as shown by Sjögren [323]; related positive weak type (1, 1) results are contained in the article of Vargas [365]. The precise value of the operator norm of the uncentered Hardy–Littlewood maximal function on was shown by Grafakos and Montgomery-Smith [136] to be the unique positive solution of the equation . This constant raised to the power n is the operator norm of the strong maximal function M s on for . The best weak type (1, 1) constant for the centered Hardy–Littlewood maximal operator was shown by Melas [248] to be the largest root of the quadratic equation 12x ² − 22x + 5 = 0. The strong maximal operator M s is not weak type (1, 1), but it satisfies the substitute inequality . This result is due to Jessen, Marcinkiewicz, and Zygmund [176], but a geometric proof of it was obtained by Córdoba and Fefferman [73].

The basic facts about the Fourier transform go back to Fourier [119]. The theory of distributions was developed by Schwartz [314], [315]. For a concise introduction to the theory of distributions we refer to Hörmander [160] and Yosida [382]. Homogeneous distributions were considered by Riesz [295] in the study of the Cauchy problem in partial differential equations, although some earlier accounts are found in the work of Hadamard. They were later systematically studied by Gelfand and Šilov [126], [127]. References on the uncertainty principle include the articles of Fefferman [114] and Folland and Sitaram [118]. The best possible constant B p in the Hausdorff–Young inequality when 1 ≤ p ≤ 2 was shown by Beckner [21] to be . This best constant was previously obtained by Babenko [16] in the case when p′ is an even integer.

A nice treatise of the spaces is found in Hörmander [159]. This reference also contains Theorem 2.5.6, which is due to him. Theorem 2.5.16 is due to de Leeuw [94], but the proof presented here is taken from Jodeit [178]. De Leeuw’s result in Exercise 2.5.9 says that periodic elements of can be isometrically identified with elements of , the latter being the space of all multipliers on . The hint in Exercise 2.5.13 was suggested by M. Peloso.

Parts (b) and (c) of Proposition 2.6.7 are due to van der Corput [364] and are referred to in the literature as van der Corput’s lemma. The refinement in part (a) was subsequently obtained by Arhipov, Karachuba, and Čubarikov [8]. The treatment of these results in the text is based on the article of Carbery, Christ, and Wright [53], which also investigates higher-dimensional analogues of the theory. Precise asymptotics can be obtained for a variety of oscillatory integrals via the method of stationary phase; see Hörmander [160]. References on oscillatory integrals include the books of Titchmarsh [362], Erdélyi [107], Zygmund [388], [389], Stein [344], and Sogge [328]. The latter provides a treatment of Fourier integral operators.

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© Springer Science+Business Media New York 2014

Loukas GrafakosClassical Fourier AnalysisGraduate Texts in Mathematics24910.1007/978-1-4939-1194-3_3

3. Fourier Series

Loukas Grafakos1 

(1)

Department of Mathematics, University of Missouri, Columbia, MO, USA

Principles of Fourier series go back to ancient times. The attempts of the Pythagorean school to explain musical harmony in terms of whole numbers embrace early elements of a trigonometric nature. The theory of epicycles in the Almagest of Ptolemy, based on work related to the circles of Appolonius, contains ideas of astronomical periodicities that we would interpret today as harmonic analysis. Early studies of acoustical and optical phenomena, as well as periodic astronomical and geophysical occurrences, provided a stimulus in the physical sciences toward the rigorous study of expansions of periodic functions. This study is carefully pursued in this chapter.

The modern theory of Fourier series begins with attempts to solve boundary value problems using trigonometric functions. The work of d’Alembert, Bernoulli, Euler, and Clairaut on the vibrating string led to the belief that it might be possible to represent arbitrary periodic functions as sums of sines and cosines. Fourier announced belief in this possibility in his solution of the problem of heat distribution in spatial bodies (in particular, for the cube $$\mathbf{T}^{3}$$) by expanding an arbitrary function of three variables as a triple sine series. Fourier’s approach, although heuristic, was appealing and eventually attracted attention. It was carefully studied and further developed by many scientists, but most notably by Laplace and Dirichlet, who were the first to investigate the validity of the representation of a function in terms of its Fourier series. This is the main topic of study in this chapter.

3.1 Fourier Coefficients

We discuss some basic facts of Fourier analysis on the torus $$\mathbf{T}^{n}$$. Throughout this chapter, n denotes the dimension, i.e., a fixed positive integer.

3.1.1 The n-Torus $$\mathbf{T}^{n}$$

The n-torus $$\mathbf{T}^{n}$$ is the cube [0, 1] n with opposite sides identified. This means that the points $$(x_{1},\ldots,0,\ldots,x_{n})$$ and $$(x_{1},\ldots,1,\ldots,x_{n})$$ are identified whenever 0 and 1 appear in the same coordinate. A more precise definition can be given as follows: We say that x, y in $$\mathbf{R}^{n}$$ are equivalent and we write

(3.1.1)

if $$x - y \in \mathbf{Z}^{n}$$. Here $$\mathbf{Z}^{n}$$ is the additive subgroup of all points in $$\mathbf{R}^{n}$$ with integer coordinates. It is a simple fact that $$\equiv $$ is an equivalence relation that partitions $$\mathbf{R}^{n}$$ into equivalence classes. The $$n$$-torus $$\mathbf{T}^{n}$$ is then defined as the set $$\mathbf{R}^{n}/\mathbf{Z}^{n}$$ of all such equivalence classes. When n = 1, this set can be geometrically viewed as a circle by bending the line segment [0, 1] so that its endpoints are brought together. When n = 2, the identification brings together the left and right sides of the unit square [0, 1]² as well as the top and bottom sides. The resulting figure is a two-dimensional manifold embedded in R ³ that looks like a donut. See Figure 3.1.

Fig. 3.1

The graph of the two-dimensional torus $$\mathbf{T}^{2}$$.

The n-torus is an additive group. The identity element of the group is 0, which of course coincides with every $$e_{j} = (0,\ldots,0,1,0,\ldots,0)$$. To avoid multiple appearances of the identity element in the group, we often think of the n-torus as the set [−1∕2, 1∕2] n . Since the group $$\mathbf{T}^{n}$$ is additive, the inverse of an element $$x \in \mathbf{T}^{n}$$ is denoted by − x. For example, − (1∕3, 1∕4) ≡ (2∕3, 3∕4) on $$\mathbf{T}^{2}$$, or, equivalently, $$-(1/3,1/4) - (2/3,3/4) \in \mathbf{Z}^{2}$$.

The n-torus $$\mathbf{T}^{n}$$ can also be thought of as the following subset of $$\mathbf{C}^{n}$$,

(3.1.2)

in a way analogous to which the unit interval [0, 1] can be thought of as the unit circle in $$\mathbf{C}$$ once 1 and 0 are identified.

Functions on $$\mathbf{T}^{n}$$ are functions f on $$\mathbf{R}^{n}$$ that satisfy f(x + m) = f(x) for all $$x \in \mathbf{R}^{n}$$ and $$m \in \mathbf{Z}^{n}$$. Such functions are called 1-periodic in every coordinate. Haar measure on the n-torus is the restriction of n-dimensional Lebesgue measure to the set $$\mathbf{T}^{n} = [0,1]^{n}$$. This measure is still denoted by dx, while the measure of a set $$A \subseteq \mathbf{T}^{n}$$ is denoted by | A | . Translation invariance of Lebesgue measure and the periodicity of functions on $$\mathbf{T}^{n}$$ imply that for all integrable functions f on $$\mathbf{T}^{n}$$, we have

(3.1.3)

for any real numbers $$a_{1},\ldots,a_{n}$$. In view of periodicity, integration by parts on the torus does not produce boundary terms; given f, g continuously differentiable functions on $$\mathbf{T}^{n}$$ we have

Elements of $$\mathbf{Z}^{n}$$ are denoted by $$m = (m_{1},\ldots,m_{n})$$. For $$m \in \mathbf{Z}^{n}$$, we define the total size of m to be the number $$\vert m\vert = (m_{1}^{2} +\ldots +m_{n}^{2})^{1/2}$$. Recall that for $$x = (x_{1},\ldots,x_{n})$$ and $$y = (y_{1},\ldots,y_{n})$$ in $$\mathbf{R}^{n}$$,

denotes the usual dot product. Finally, for $$x \in \mathbf{T}^{n}$$, | x | denotes the usual Euclidean norm of x. If we identify $$\mathbf{T}^{n}$$ with [−1∕2, 1∕2] n , then | x | can be interpreted as the distance of the element x from the origin, and then we have $$0 \leq \vert x\vert \leq \sqrt{n}/2$$ for all $$x \in \mathbf{T}^{n}$$.

Multi-indices are elements of $$(\mathbf{Z}^{+} \cup \{ 0\})^{n}$$. For a multi-index $$\alpha = (\alpha _{1},\ldots,\alpha _{n})$$, we denote the partial derivative $$\partial _{1}^{\alpha _{1}}\cdots \partial _{n}^{\alpha _{n}}f$$ by $$\partial ^{\alpha }f$$. The spaces $$\mathcal{C}^{k}(\mathbf{T}^{n})$$ of continuously differentiable functions of order k , where $$k \in \mathbf{Z}^{+}$$, are defined as the sets of functions $$\varphi$$ for which $$\partial ^{\alpha }\varphi$$ exist and are continuous for all $$\vert \alpha \vert \leq k$$. When k = 0 we set $$\mathcal{C}^{0}(\mathbf{T}^{n}) = \mathcal{C}(\mathbf{T}^{n})$$ to be the space of continuous functions on $$\mathbf{T}^{n}$$. The space $$\mathcal{C}^{\infty }(\mathbf{T}^{n})$$ of infinitely differentiable functions on $$\mathbf{T}^{n}$$ is the union of all the $$\mathcal{C}^{k}(\mathbf{T}^{n})$$. All of these spaces are contained in $$L^{p}(\mathbf{T}^{n})$$, which are nested, with $$L^{1}(\mathbf{T}^{n})$$ being the largest.

3.1.2 Fourier Coefficients

Definition 3.1.1.

For a complex-valued function f in $$L^{1}(\mathbf{T}^{n})$$ and $$m$$ in $$\mathbf{Z}^{n}$$, we define

(3.1.4)

We call $$\widehat{f}(m)$$ the mth Fourier coefficient of f. We note that $$\widehat{f}(\xi )$$ is not defined for $$\xi \in \mathbf{R}^{n}\setminus \mathbf{Z}^{n}$$, since the function $$x\mapsto e^{-2\pi i\xi \cdot x}$$ is not 1-periodic in any coordinate and therefore not well defined on $$\mathbf{T}^{n}$$. For a finite Borel measure μ on $$\mathbf{T}^{n}$$ and $$m \in \mathbf{Z}^{n}$$ the expression

(3.1.5)

is called the mth Fourier coefficient of μ.

The Fourier series of f at $$x \in \mathbf{T}^{n}$$ is the series

(3.1.6)

It is not clear at present in which sense and for which $$x \in \mathbf{T}^{n}$$ (3.1.6) converges. The study of convergence of Fourier series is the main topic of study in this chapter.

We quickly recall the notation we introduced in Chapter 2.​ We denote by $$\overline{f}$$ the complex conjugate of the function f, by $$\widetilde{f}$$ the function $$\widetilde{f}(x) = f(-x)$$, and by τ y (f) the function $$\tau ^{y}(f)(x) = f(x - y)$$ for all $$y \in \mathbf{T}^{n}$$. We mention some elementary properties of Fourier coefficients.

Proposition 3.1.2.

Let f, g be in $$L^{1}(\mathbf{T}^{n})$$ . Then for all $$m,k \in \mathbf{Z}^{n}$$, $$\lambda \in \mathbf{C}$$, $$y \in \mathbf{T}^{n}$$ , and all multi-indices $$\alpha$$ we have

(1)

$$\,\,\,\widehat{f + g}(m) =\widehat{ f}(m) +\widehat{ g}(m)$$,

(2)

$$\,\,\,\widehat{\lambda f}(m) =\lambda \widehat{ f}(m)$$,

(3)

$$\,\,\,\widehat{\overline{f}\,}(m) = \overline{\widehat{f}(-m)}$$,

(4)

$$\,\,\,\widehat{\widetilde{f}\,\,}(m) =\widehat{ f}(-m)$$,

(5)

$$\,\,\,\widehat{\tau ^{y}(f)}(m) =\widehat{ f}(m)e^{-2\pi im\cdot y}$$,

(6)

$$\,\,\,(e^{2\pi ik(\cdot )}f)\,\,\widehat{\,}\,\,(m) =\widehat{ f}(m - k)$$,

(7)

$$\,\,\,\widehat{f}(0) =\int _{\mathbf{T}^{n}}f(x)\,dx$$,

(8)

$$\,\,\,\sup \limits _{m\in \mathbf{Z}^{n}}\vert \widehat{f}(m)\vert \leq \big\| f\big\|_{L^{1}(\mathbf{T}^{n})}$$,

(9)

$$\,\,\,\widehat{f {\ast} g}(m) =\widehat{ f}(m)\widehat{g}(m)$$,

(10)

$$\,\,\,\widehat{\partial ^{\alpha }f}(m) = (2\pi im)^{\alpha }\widehat{f}(m)$$ , whenever $$f \in \mathcal{C}^{\alpha }$$ .

Proof.

The proof of properties (1)–(10) is rather easy and is left to the reader. We only sketch the proof of (9). We have

where the interchange of integrals is justified by the absolute convergence of the integrals and Fubini’s theorem. □ 

Remark 3.1.3.

The Fourier coefficients have the following property. For a function f ₁ on $$\mathbf{T}^{n_{1}}$$ and a function f ₂ on $$\mathbf{T}^{n_{2}}$$, the tensor function

is a periodic function on $$\mathbf{T}^{n_{1}+n_{2}}$$ whose Fourier coefficients are

(3.1.7)

for all $$m_{1} \in \mathbf{Z}^{n_{1}}$$ and $$m_{2} \in \mathbf{Z}^{n_{2}}$$.

Definition 3.1.4.

A trigonometric polynomial on $$\mathbf{T}^{n}$$ is a function of the form

(3.1.8)

where $$\{a_{m}\}_{m\in \mathbf{Z}^{n}}$$ is a finitely supported sequence in $$\mathbf{Z}^{n}$$. The degree of P is the largest number $$\vert q_{1}\vert +\ldots +\vert q_{n}\vert $$ such that a q is nonzero, where $$q = (q_{1},\ldots,q_{n})$$. Observe that in view of the orthonormality of the exponentials we have for all $$m \in \mathbf{Z}^{n}$$

Example 3.1.5.

If the sequence $$\{a_{m}\}_{m}$$ has only one nonzero term, then the trigonometric polynomial of Definition 3.1.4 reduces to a trigonometric monomial , which has the form

for some $$q = (q_{1},\ldots,q_{n}) \in \mathbf{Z}^{n}$$ and $$a \in \mathbf{C}$$.

Let

be a trigonometric polynomial on $$\mathbf{T}^{n}$$ and let μ be a finite Borel measure on $$\mathbf{T}^{n}$$. Then we have

(3.1.9)

In particular, if f is an integrable function on $$\mathbf{T}^{n}$$ we have

(3.1.10)

This implies that the partial sums

of the Fourier series of f in (3.1.6) can be obtained by convolving f with the function

(3.1.11)

This function is called the Dirichlet kernel.

3.1.3 The Dirichlet and Fejér Kernels

Definition 3.1.6.

Let $$0 \leq R < \infty $$. The square Dirichlet kernel on $$\mathbf{T}^{n}$$ is the function

(3.1.12)

The circular (or spherical) Dirichlet kernel on $$\mathbf{T}^{n}$$ is the function

(3.1.13)

In dimension n = 1 these functions coincide and are denoted by

This function is called the Dirichlet kernel and coincides with D N (x) in (3.1.11) when $$N \leq R < N + 1$$ and $$N \in \mathbf{Z}^{+} \cup \{ 0\}$$; see Figure 3.2.

Fig. 3.2

The graph of the Dirichlet kernel D ₅ plotted on the interval [−1∕2, 1∕2].

Both the square and circular (or spherical) Dirichlet kernels are trigonometric polynomials. The square Dirichlet kernel on $$\mathbf{T}^{n}$$ is equal to a product of one-dimensional Dirichlet kernels, that is,

(3.1.14)

We have the following two equivalent ways to write the Dirichlet kernel D N :

(3.1.15)

for x ∈ [0, 1]. To verify the validity of (3.1.15), we write

It follows that for $$R \in \mathbf{R}^{+} \cup \{ 0\}$$ we have

(3.1.16)

It is reasonable to ask whether the family $$\{D_{R}\}_{R>0}$$ forms an approximate identity as $$R \rightarrow \infty $$. Using (3.1.15) we see that each D R is integrable over [−1∕2, 1∕2] and has integral equal to 1. But it follows from Exercise 3.1.5 that $$\|D_{R}\|_{L^{1}} \approx \log R$$ as $$R \rightarrow \infty $$, and therefore property (i) in Definition 1.​2.​15 fails for D R . We conclude that the family $$\{D_{R}\}_{R>0}$$ is not an approximate identity on $$\mathbf{T}^{1}$$, which significantly complicates the study of Fourier series. Consequently, the family $$\{D_{R}^{n}\}_{R>0}$$ is not an approximate identity on $$\mathbf{T}^{n}$$, since $$\|D_{R}^{n}\|_{L^{1}(\mathbf{T}^{1})} \approx (\log R)^{n}$$. The same is true for the family of circular (or spherical) Dirichlet kernels $$\{\stackrel{\circ }{D_{R}^{n}}\}_{R>0}$$. Although this is harder to prove, it will be a consequence of the results in Section 4.​2.​

A typical situation encountered in analysis is that the means of a sequence behave better than the original sequence. This fact led Cesàro and independently Fejér to consider the arithmetic means of the Dirichlet kernel in dimension 1, that is, the expressions

(3.1.17)

The expression in (3.1.17) is in fact equal to the Fejér kernel given in Example 1.​2.​18.​ We have the following identity concerning the kernel F N .

Proposition 3.1.7.

For every nonnegative integer N the identity holds

(3.1.18)

for all $$x \in \mathbf{T}^{1}$$ . Thus $$\widehat{F_{N}}(m) = 1 - \tfrac{\vert m\vert } {N+1}$$ if $$\vert m\vert \leq N$$ and zero otherwise.

Proof.

The fact that the expression in (3.1.17) is equal to the middle term in (3.1.18) is a consequence of the trivial calculation:

To verify the second equality in (3.1.18) we use the simple geometric series identity $$1 + r + r^{2} + \cdots + r^{N} = \frac{1-r^{N+1}} {1-r}$$ to write for x ≠ 0

from which it follows that

(3.1.19)

Likewise we prove that

(3.1.20)

Adding (3.1.19) and (3.1.20) we deduce

(3.1.21)

Multiplying (3.1.21) by $$- \frac{1} {N+1}$$ and adding D N (x) we obtain

Writing the preceding expression on the right as

computing the derivative of the expression in the curly brackets, and simplifying, we finally obtain that

(3.1.22)

This proves the second identity in (3.1.18). □ 

Definition 3.1.8.

Let N be a nonnegative integer. The function F N on $$\mathbf{T}^{1}$$ given by (3.1.22) is called the Fejér kernel .

The Fejér kernel F N n on $$\mathbf{T}^{n}$$ is defined as the product of the 1-dimensional Fejér kernels, or as the average of the product of the Dirichlet kernels in each variable, precisely, $$F_{N}^{1}(x) = F_{N}(x)$$ and

Note that $$F_{N}^{n}$$ is a trigonometric polynomial of degree nN.

Remark 3.1.9.

Using the identities for F N in (3.1.18), we may write for all N ≥ 0

(3.1.23)

(3.1.24)

thus $$F_{N}^{n} \geq 0$$. Observe that $$F_{0}^{n}(x) = 1$$ for all $$x \in \mathbf{T}^{n}$$ and that $$F_{N}^{n}(0) = (N + 1)^{n}$$.

Proposition 3.1.10.

The family of Fejér kernels $$\{F_{N}^{n}\}_{N=0}^{\infty }$$ is an approximate identity on $$\mathbf{T}^{n}$$ .

Proof.

Since $$F_{N}^{n} \geq 0$$ we have that $$\|F_{N}^{n}\|_{L^{1}} =\int _{\mathbf{T}^{n}}F_{N}^{n}\,dx$$. Also $$\int _{\mathbf{T}^{n}}F_{N}^{n}\,dx = 1$$, in view of identity (3.1.23). Thus properties (i) and (ii) of approximate identities (according to Definition 1.​2.​15) hold. To prove property (iii) of the definition we make use of identity (3.1.24). Using the fact that $$1 \leq \frac{\vert t\vert } {\vert \sin t\vert } \leq \frac{\pi } {2}$$ when $$\vert t\vert \leq \frac{\pi } {2}$$, we obtain

when $$\vert x\vert \leq \frac{1} {2}$$. This implies that for $$\delta > 0$$ we have

as $$N \rightarrow \infty $$. In higher dimensions, given $$x = (x_{1},\ldots,x_{n}) \in [-1/2,1/2]^{n}$$ with $$\vert x\vert \geq \delta$$, there is a $$j \in \{ 1,\ldots,n\}$$ such that $$\vert x_{j}\vert \geq \delta /\sqrt{n}$$ and thus

This proves the claim. □ 

Exercises

3.1.1. Identify $$\mathbf{T}^{1}$$ with [−1∕2, 1∕2] and let h(t) be an integrable function on $$\mathbf{T}^{1}$$.

(a) If h(t) ≥ 0 is even, show that $$\widehat{h}(m)$$ is real and $$\vert \widehat{h}(m)\vert \leq \widehat{ h}(0)$$ for all $$m \in \mathbf{Z}$$.

(b) If h(t) is odd and h(t) ≥ 0 on [0, 1∕2), then $$i\,\widehat{h}(m)$$ is real and $$\vert \widehat{h}(m)\vert \leq i\,m\,\widehat{h}(1)$$ for all $$m \in \mathbf{Z}$$.

3.1.2. Suppose that h is a periodic integrable function on [−1∕2, 1∕2) with integral zero. Define another periodic function H on $$\mathbf{T}^{1}$$ by setting

Compute $$\widehat{H}(m)$$ in terms of $$\widehat{h}(m)$$ for $$m \in \mathbf{Z}$$.

3.1.3. Suppose that $$\{g_{\varepsilon }\}_{\varepsilon >0}$$ is an approximate identity on $$\mathbf{R}^{n}$$ as $$\varepsilon \rightarrow 0$$ and let

Show that the family $$\{G_{\varepsilon }\}_{\varepsilon >0}$$ is an approximate identity on $$\mathbf{T}^{n}$$.

3.1.4. On $$\mathbf{T}^{1}$$ define the de la Vallée Poussin kernel

(a) Show that the sequence V N is an approximate identity.

(b) Prove that $$\widehat{V _{N}}(m) = 1$$ when | m | ≤ N + 1, and $$\widehat{V _{N}}(m) = 0$$ when $$\vert m\vert \geq 2N + 2$$.

3.1.5. (a) Show that for all $$\vert t\vert \leq \frac{\pi } {2}$$ we have

(b) Let $$D_{N}$$ be the Dirichlet kernel on $$\mathbf{T}^{1}$$. Prove that for $$N \in \mathbf{Z}^{+}$$ we have

Conclude that the numbers $$\|D_{N}\|_{L^{1}}$$ grow logarithmically as $$N \rightarrow \infty $$ and therefore the family $$\{D_{N}\}_{N=1}^{\infty }$$ is not an approximate identity on $$\mathbf{T}^{1}$$. The numbers $$\|D_{N}\|_{L^{1}}$$, $$N = 1,2,\ldots$$ are called the Lebesgue constants .

[Hint: Part (a): Show that the derivative of $$\tfrac{1} {\sin (t)} -\tfrac{1} {t}$$ is nonnegative on $$(0, \frac{\pi } {2}]$$, or equivalently prove that $$\tan (t)\sin (t) \geq t^{2}$$ on $$(0, \frac{\pi } {2}]$$; this is a consequence of the inequality $$\sqrt{\sin (t)\tan (t)} \geq 2( \frac{1} {\sin (t)} + \frac{1} {\tan (t)})^{-1} = 2\tan ( \frac{t} {2}) \geq t$$. Part (b): Replace D N (t) by $$\frac{\sin ((2N+1)\pi t)} {\pi t}$$ and estimate the difference using part (a).]

3.1.6. Let D N be the Dirichlet kernel on $$\mathbf{T}^{1}$$. Prove that for all $$1 < p < \infty $$ there exist two constants $$C_{p},c_{p} > 0$$ such that

[Hint: Consider the two closest zeros of D N near the origin and split the integral into the intervals thus obtained.]

3.1.7.

The Poisson kernel on $$\mathbf{T}^{n}$$ is the function

and is defined for $$0 < r_{1},\ldots,r_{n} < 1$$. Prove that $$P_{r_{1},\ldots,r_{n}}$$ can be written as

and conclude that $$P_{r,\ldots,r}(x)$$ is an approximate identity as r ↑ 1.

3.2 Reproduction of Functions from Their Fourier Coefficients

We can obtain very interesting results using the Fejér kernel.

Proposition 3.2.1.

The set of trigonometric polynomials is dense in $$L^{p}(\mathbf{T}^{n})$$ for $$1 \leq p < \infty $$ .

Proof.

Given f in $$L^{p}(\mathbf{T}^{n})$$ for $$1 \leq p < \infty $$, consider $$f {\ast} F_{N}^{n}$$. Clearly $$f {\ast} F_{N}^{n}$$ is also a trigonometric polynomial. In view of Theorem 1.​2.​19 (1), $$f {\ast} F_{N}^{n}$$ converges to f in $$L^{p}$$ as $$N \rightarrow \infty $$. □ 

Corollary 3.2.2.

(Weierstrass approximation theorem for trigonometric polynomials) Every continuous function on the torus is a uniform limit of trigonometric polynomials.

Proof.

Since f is continuous on $$\mathbf{T}^{n}$$, which is a compact set, Theorem 1.​2.​19 (2) gives that $$f {\ast} F_{N}^{n}$$ converges uniformly to f as $$N \rightarrow \infty $$. But $$f {\ast} F_{N}^{n}$$ is a trigonometric polynomial, and so we conclude that every continuous function on $$\mathbf{T}^{n}$$ can be uniformly approximated by trigonometric polynomials. □ 

3.2.1 Partial sums and Fourier inversion

We now define the partial sums of Fourier series.

Definition 3.2.3.

Let $$R \geq 0$$ and $$N \in \mathbf{Z}^{+} \cup \{ 0\}$$. The expressions

are called the square partial sums of the Fourier series of f. Then the expressions

are called the circular (or spherical) partial sums of the Fourier series of f. The expressions

are called the square Cesàro means (or square Fejér means) of f.

Finally, for R ≥ 0 the expressions

are called the circular Cesàro means (or circular Fejér means) of f.

Observe that $$f {\ast}\stackrel{\circ }{F_{R}^{n}}$$ is equal to the average of the expressions $$f {\ast}\stackrel{\circ }{D_{r}^{n}}$$ in the following sense:

This is analogous to the fact that the Fejér kernel F N is the average of the Dirichlet kernels D ₀, $$D_{1},\ldots,D_{N}$$.

A fundamental problem is in what sense the partial sums of the Fourier series converge back to the function as $$R \rightarrow \infty $$ or $$N \rightarrow \infty $$. This problem is of central importance in harmonic analysis and is in part investigated in this chapter.

We now ask the question whether the Fourier coefficients uniquely determine the function. The answer is affirmative and simple.

Proposition 3.2.4.

If $$f,g \in L^{1}(\mathbf{T}^{n})$$ satisfy $$\widehat{f}(m) =\widehat{ g}(m)$$ for all m in $$\mathbf{Z}^{n}$$ , then f = g a.e.

Proof.

By linearity of the problem, it suffices to assume that g = 0. If $$\widehat{f}(m) = 0$$ for all $$m \in \mathbf{Z}^{n}$$, Definition 3.2.3 implies that $$F_{N}^{n} {\ast} f = 0$$ for all $$N \in \mathbf{Z}^{+}$$. In view of Proposition 3.1.10, the sequence $$\{F_{N}^{n}\}_{N\in \mathbf{Z}^{+}}$$ is an approximate identity as $$N \rightarrow \infty $$. Therefore,

as $$N \rightarrow \infty $$; hence $$\|f\|_{L^{1}} = 0$$, from which we conclude that f = 0 a.e. □ 

A useful consequence of the result just proved is the following.

Proposition 3.2.5.

(Fourier inversion) Suppose that $$f \in L^{1}(\mathbf{T}^{n})$$ and that

Then

(3.2.1)

and therefore f is almost everywhere equal to a continuous function.

Proof.

It is straightforward to check that both functions in (3.2.1) are well defined and have the same Fourier coefficients. Therefore, they must be almost everywhere equal by Proposition 3.2.4. Moreover, the function on the right in (3.2.1) is everywhere continuous. □ 

3.2.2 Fourier series of square summable functions

Let H be a separable Hilbert space with complex inner product $$\langle \cdot \,\vert \,\cdot \rangle$$. Recall that a subset E of H is called orthonormal if 〈f | g〉 = 0 for all f, g in E with f ≠ g, while $$\langle f\,\vert \,f\rangle = 1$$ for all f in E. A complete orthonormal system is a subset of H having the additional property that the only vector orthogonal to all of its elements is the zero vector. We summarize basic properties about orthonormal systems in the proposition below (see [307]).

Proposition 3.2.6.

Let H be a separable Hilbert space and let $$\{\varphi _{k}\}_{\!k\in \mathbf{Z}}$$ be an orthonormal system in H. Then the following are equivalent: (1) $$\{\varphi _{k}\}_{k\in \mathbf{Z}}$$ is a complete orthonormal system. (2) For every $$f \in H$$ we have

 (3) For every f ∈ H we have

where the series converges in H.

Now consider the Hilbert space space $$L^{2}(\mathbf{T}^{n})$$ with inner product

Let $$\varphi _{m}$$ be the sequence of functions $$\xi \mapsto e^{2\pi im\cdot \xi }$$ indexed by $$m \in \mathbf{Z}^{n}$$. The orthonormality of the sequence $$\{\varphi _{m}\}$$ is a consequence of the following simple but crucial identity:

The completeness of the sequence $$\{\varphi _{m}\}$$ is also evident. Since $$\big\langle f\,\vert \,\varphi _{m}\big\rangle =\widehat{ f}(m)$$ for all $$f \in L^{2}(\mathbf{T}^{n})$$, it follows from Proposition 3.2.4 that if $$\big\langle f\,\vert \,\varphi _{m}\big\rangle = 0$$ for all $$m \in \mathbf{Z}^{n}$$, then f = 0 a.e.

The next result is a consequence of Proposition 3.2.6.

Proposition 3.2.7.

The following are valid for $$f,g \in L^{2}(\mathbf{T}^{n})$$:  (1) (Plancherel’s identity)

 (2) The function f(t) is a.e. equal to the $$L^{2}(\mathbf{T}^{n})$$ limit of the sequence

 (3) (Parseval’s relation)

 (4) The map $$f\mapsto \{\widehat{f}(m)\}_{m\in \mathbf{Z}^{n}}$$ is an isometry from $$L^{2}(\mathbf{T}^{n})$$ onto ℓ ² . (5) For all $$k \in \mathbf{Z}^{n}$$ we have

Proof.

(1) and (2) follow from the corresponding statements in Proposition 3.2.6. Notice that both sides of (3) converge by the Cauchy-Schwarz inequality. Parseval’s relation (3) follows from polarization . By this we mean the following procedure. First replace f by f + g in (1) and expand the squares to obtain

and from this it follows that the real parts of the expressions in (3) are equal. Next replace f by f + ig in (1) and expand the squares. Using $$\mbox{ Re}\,(-iw) = \mbox{ Im}\,w$$ we obtain

and thus the imaginary parts of the expressions in (3) are equal. Thus (3) holds. Next we prove (4). We already know that the map $$f\mapsto \{\widehat{f}(m)\}_{m\in \mathbf{Z}^{n}}$$ is an injective isometry. It remains to show that it is onto. Given a square summable sequence $$\{a_{m}\}_{m\in \mathbf{Z}^{n}}$$ of complex numbers, define

Observe that f N is a Cauchy sequence in $$L^{2}(\mathbf{T}^{n})$$ and it therefore converges to some $$f \in L^{2}(\mathbf{T}^{n})$$. Then we have $$\widehat{f}(m) = a_{m}$$ for all $$m \in \mathbf{Z}^{n}$$. Finally, (5) is a consequence of (3) and Proposition 3.1.2 (6) and (3). □ 

3.2.3 The Poisson Summation Formula

We end this section with an important result that connects Fourier analysis on the torus with Fourier analysis on $$\mathbf{R}^{n}$$. Suppose that f is an integrable function on $$\mathbf{R}^{n}$$ and let $$\widehat{f}$$ be its Fourier transform. Restrict $$\widehat{f}$$ on $$\mathbf{Z}^{n}$$ and form the “Fourier series” (assuming that it converges)

What does this series represent? Since the preceding function is 1-periodic in every variable, it follows that it cannot be equal to f, unless f is itself periodic. However, it should not come as a surprise that it is in fact equal to the periodization of f on $$\mathbf{R}^{n}$$. In other words, the Fourier expansion of a function on $$\mathbf{R}^{n}$$ reproduces the periodization of the function.

Theorem 3.2.8.

(Poisson summation formula) Let f be a continuous function on $$\mathbf{R}^{n}$$ which satisfies for some $$C,\delta > 0$$ and for all $$x \in \mathbf{R}^{n}$$

and whose Fourier transform $$\widehat{f}$$ restricted on $$\mathbf{Z}^{n}$$ satisfies

(3.2.2)

Then for all $$x \in \mathbf{R}^{n}$$ we have

(3.2.3)

and in particular

Proof.

Define a 1-periodic function on $$\mathbf{T}^{n}$$ by setting

It is straightforward to verify that $$\|F\|_{L^{1}([0,1]^{n})} =\| f\|_{L^{1}(\mathbf{R}^{n})}$$, thus F lies in $$L^{1}(\mathbf{T}^{n})$$. We prove that the sequence of the Fourier coefficients of F coincides with the restriction of the Fourier transform of f on $$\mathbf{Z}^{n}$$. This follows from the calculation

in which the interchange of the sum and the integral is justified by the Weierstrass M-test of uniform convergence of series, since

where we used $$\vert k + x\vert \geq \vert k\vert -\vert x\vert \geq \vert k\vert -\sqrt{n}$$. This calculation also shows that F is the sum of a uniformly convergent series of continuous functions on [0, 1] n , thus it is itself continuous. It follows that (3.2.2) holds with $$\vert \widehat{F}(m)\vert $$ in place of $$\vert \widehat{f}(m)\vert $$. Hence, Proposition 3.2.5 applies, and given the fact that F is continuous, it yields conclusion (3.2.3) for all $$x \in \mathbf{T}^{n}$$ and, by periodicity, for all $$x \in \mathbf{R}^{n}$$. □ 

Example 3.2.9.

We have seen earlier (Exercise 2.2.11) that the following identity gives the Fourier transform of the Poisson kernel in $$\mathbf{R}^{n}$$:

The Poisson summation formula yields the identity

(3.2.4)

which implies

(3.2.5)

It follows from (3.2.4) that

from which we obtain the identity

(3.2.6)

The limit in (3.2.6) can be easily calculated in dimension 1 using that

and this yields

Also, in dimension 1, from (3.2.5) we obtain the related identity

Example 3.2.10.

Let $$0 < \mbox{ Re }\alpha < n$$. We introduce a smooth function $$\widehat{\varPhi }(\xi )$$ which is equal to 1 on the ball $$\vert \xi \vert \leq 1$$ and vanishes outside the ball $$\vert \xi \vert \leq 2$$. We investigate the behavior as x → 0 of the expression

when $$x \in [-\tfrac{1} {2}, \tfrac{1} {2})^{n}\setminus \{0\}$$. As in Example 2.​4.​9, let η be a smooth radial function on $$\mathbf{R}^{n}$$ that is equal to 1 outside the ball B(0, 1∕2) and vanishes on the ball B(0, 1∕4) and define

Let $$\varPhi _{\delta }(x) =\delta ^{-n}\varPhi (x/\delta )$$. The Poisson summation formula (Theorem 3.2.8) gives

It was shown in Example 2.​4.​9 that g(y) decays faster than the reciprocal of any polynomial at infinity and is equal to $$\pi ^{\alpha -\frac{n} {2} }\varGamma (\frac{n-\alpha } {2})\varGamma ( \frac{\alpha }{2})^{-1}\vert y\vert ^{\alpha -n} + h(y)$$, where h is a smooth function on $$\mathbf{R}^{n}$$. Since x ≠ 0, it follows that g is smooth in a small relatively compact neighborhood of x and, since $$\{\varPhi _{1/R}\}_{R>0}$$ is an approximate identity, Theorem 1.​2.​19 (2) yields that $$(g {\ast}\varPhi _{1/R})(x) \rightarrow g(x)$$ as $$R \rightarrow \infty $$. Assume for a moment that

(3.2.7)

Since x + m does not vanish for any $$m \in \mathbf{Z}^{n}\setminus \{0\}$$, the function g is smooth in a relatively compact neighborhood of x + m, and thus $$(g {\ast}\varPhi _{1/R})(x + m) \rightarrow g(x + m)$$ as $$R \rightarrow \infty $$. Consequently, the sum on the right in (3.2.7) is equal to

We conclude that

where h ₁ is a $$\mathcal{C}^{\infty }$$ function on $$[-\tfrac{1} {2}, \tfrac{1} {2})^{n}$$ given by

We now explain the passage of the limit inside the sum in (3.2.7). This is a consequence of the Lebesgue dominated convergence theorem, provided we know that

(3.2.8)

for some constant C independent of R ≥ 1 and of m. Indeed, the expression on the left of this inequality is bounded by I + II, where

In I we have

hence

In II we use

while the term left produces a convergent integral, which is uniformly bounded in $$R \geq 1$$. This proves (3.2.8).

Exercises

3.2.1. On $$\mathbf{T}^{1}$$ let P be a trigonometric polynomial of degree N > 0. Show that P has at most 2N zeros. Construct a trigonometric polynomial with exactly 2N zeros.

3.2.2. (Hausdorff–Young inequality) Prove that when $$f \in L^{p}(\mathbf{T}^{n})$$, $$1 \leq p \leq 2$$, the sequence of Fourier coefficients of f is in $$\ell^{p'}(\mathbf{Z}^{n})$$ and

Also observe that 1 is the best constant in the preceding inequality.

3.2.3. Use without proof that there exists a constant C > 0 such that for all $$t \in \mathbf{R}$$ we have

to construct an example of a continuous function g on $$\mathbf{T}^{1}$$ with

for all q < 2. Thus the Hausdorff–Young inequality of Exercise 3.2.2 fails for p > 2.

[Hint: Consider $$g(x) =\sum _{ k=2}^{\infty } \frac{e^{ik\mathrm{log}k}} {k^{1/2}(\mathrm{log}k)^{2}} e^{2\pi ikx}$$. For a proof of the previous estimate, see Zygmund [388, Theorem (4.7) p. 199].]

3.2.4. (S. Bernstein) Let P(x) be a trigonometric polynomial of degree N on $$\mathbf{T}^{1}$$. Prove that $$\|P'\|_{L^{\infty }} \leq 4\pi N\|P\|_{L^{\infty }}$$.

[Hint: Prove first that $$P'(x)/2\pi iN$$ is equal to

and then take $$L^{\infty }$$ norms.]

3.2.5. (Fejér and F. Riesz) Let $$P(\xi ) =\sum _{ k=-N}^{N}a_{k}e^{2\pi ik\xi }$$ be a trigonometric polynomial on $$\mathbf{T}^{1}$$ of degree N such that P(ξ) > 0 for all ξ. Prove that there exists a trigonometric polynomial Q(ξ) of the form $$\sum _{k=0}^{N}b_{k}e^{2\pi ik\xi }$$ such that $$P(\xi ) = \vert Q(\xi )\vert ^{2}$$.

[Hint: Since P ≥ 0 the complex-variable polynomial $$R(z) =\sum _{ k=-N}^{N}a_{k}z^{k+N}$$ must satisfy $$R(z) = z^{2N}\overline{R(1/\overline{z})}$$, and thus it must have N zeros inside the unit circle and the other N outside. Therefore we may write $$R(z) = a_{N}\prod _{k=1}^{s}(z - z_{k})^{r_{k}}(z - 1/\overline{z_{ k}})^{r_{k}}$$ for some 0 <  | z k  |  < 1 and r k  ≥ 1 with $$\sum _{k=1}^{s}r_{k} = N$$. Then take $$z = e^{2\pi i\xi }$$.]

3.2.6. Let g be a function on $$\mathbf{R}^{n}$$ that satisfies $$\vert g(x)\vert + \vert \widehat{g}(x)\vert \leq C(1 + \vert x\vert )^{-n-\delta }$$ for some $$C,\delta > 0$$ and all $$x \in \mathbf{R}^{n}$$. Prove that

for any $$x,\alpha \in \mathbf{R}^{n}$$ and $$\lambda > 0$$.

3.2.7. Verify the following identity when 0 < r < 1 and $$x \in \mathbf{R}^{n}$$

In the special case n = 1 and $$x \in \mathbf{R}$$ we have

[Hint: Use identity (3.2.4) and Exercise 3.1.7 when n = 1.]

3.2.8.

Let $$\gamma \in \mathbf{R}$$ and $$\lambda > 0$$. Show that

[Hint: Use Exercise 3.2.6 when n = 1 with x = 0, $$\alpha = -\gamma \lambda$$, $$g(x) = \frac{1} {\pi } \frac{1} {1+x^{2}}$$ and sum in m.]

3.3 Decay of Fourier Coefficients

In this section we investigate the interplay between the smoothness of a function and the decay of its Fourier coefficients.

3.3.1 Decay of Fourier Coefficients of Arbitrary Integrable Functions

We begin with the classical result asserting that the Fourier coefficients of any integrable function tend to zero at infinity. One should compare the following proposition with Proposition 2.​2.​17.​

Proposition 3.3.1.

(Riemann–Lebesgue lemma) Given a function f in $$L^{1}(\mathbf{T}^{n})$$ , we have that $$\vert \widehat{f}(m)\vert \rightarrow 0$$ as $$\vert m\vert \rightarrow \infty $$ .

Proof.

Given $$f \in L^{1}(\mathbf{T}^{n})$$ and $$\varepsilon > 0$$, let P be a trigonometric polynomial such that $$\|f - P\|_{L^{1}} <\varepsilon$$. If $$\vert m\vert > \mbox{ degree}\,(P)$$, then $$\widehat{P}(m) = 0$$ and thus

This proves that $$\vert \widehat{f}(m)\vert \rightarrow 0$$ as $$\vert m\vert \rightarrow \infty $$. □ 

Several questions are naturally raised. How fast can the Fourier coefficients of an L ¹ function tend to zero? Does additional smoothness of the function imply faster decay of the Fourier coefficients? Can such a decay be quantitatively expressed in terms of the smoothness of the function?

We answer the first question. Fourier coefficients of an L ¹ function may tend to zero arbitrarily slowly, that is, slower than any given rate of decay. To achieve this, we need the following two lemmas.

Lemma 3.3.2.

Given a sequence of positive real numbers $$\{a_{m}\}_{m=0}^{\infty }$$ that tends to zero as $$m \rightarrow \infty $$ , there exists a sequence $$\{c_{m}\}_{m=0}^{\infty }$$ that satisfies

(3.3.1)

for all $$m = 0,1,\ldots$$ . A sequence $$\{c_{m}\}_{m=0}^{\infty }$$ that satisfies (3.3.1) is called convex .

Proof.

Let k ₀ = 0 and suppose that a m  ≤ M for all m ≥ 0. Find $$k_{1} > k_{0}$$ such that for m ≥ k ₁ we have a m  ≤ M∕2. Now find $$k_{2} > k_{1} + \frac{k_{1}-k_{0}} {2}$$ such that for m ≥ k ₂ we have a m  ≤ M∕4. Next find $$k_{3} > k_{2} + \frac{k_{2}-k_{1}} {2}$$ such that for m ≥ k ₃ we have a m  ≤ M∕8. Continue inductively in this way and construct a subsequence $$k_{0} < k_{1} < k_{2} < \cdots $$ of the integers such that for m ≥ k j+1 we have $$a_{m} \leq 2^{-j-1}M$$ and $$k_{j+1} > k_{j} + \frac{k_{j}-k_{j-1}} {2}$$ for j ≥ 1. Join the points (k ₀, 2M), (k ₁, M), (k ₂, M∕2), $$(k_{3},M/4),\ldots$$ by straight lines and note that by the choice of the sequence $$\{k_{j}\}_{j=0}^{\infty }$$ the resulting piecewise linear function h is convex on $$[0,\infty )$$. Define c m  = h(m) and observe that the sequence $$\{c_{m}\}_{m=0}^{\infty }$$ satisfies the required properties. Exercise 3.3.1 contains an alternative proof.

 □ 

Lemma 3.3.3.

Given a convex decreasing sequence $$\{c_{m}\}_{m=0}^{\infty }$$ of positive real numbers satisfying $$\lim _{m\rightarrow \infty }c_{m} = 0$$ and a fixed integer s ≥ 0, we have that

(3.3.2)

Proof.

We begin by observing the validity of the telescoping sum

(3.3.3)

To show that the last expression tends to c s as $$N \rightarrow \infty $$, we take $$M = [\frac{N} {2} ]$$ and we use convexity ($$c_{s+M+j}\! -\! c_{s+M+j+1} \geq c_{s+M+j+1}\! -\! c_{s+M+j+2}$$) to obtain

The preceding calculation implies that $$(N + 1)(c_{s+N+1} - c_{s+N+2})$$ tends to zero as $$N \rightarrow \infty $$ and thus the expression in (3.3.3) converges to c s as $$N \rightarrow \infty $$. □ 

The proof of Lemma 3.3.3 appears more natural after examining Exercise 3.3.3(a). We now state the theorem we alluded to earlier.

Theorem 3.3.4.

Let $$(d_{m})_{m\in \mathbf{Z}^{n}}$$ be a sequence of positive real numbers with d m → 0 as $$\vert m\vert \rightarrow \infty $$ . Then there exists a function $$f \in L^{1}(\mathbf{T}^{n})$$ such that $$\widehat{f}(m) \geq d_{m}$$ for all $$m \in \mathbf{Z}^{n}$$ . In other words, given any rate of decay, there exists an integrable function on the torus whose Fourier coefficients have slower rate of decay.

Proof.

We are given a sequence of positive numbers $$\{a_{m}\}_{m\in \mathbf{Z}}$$ that converges to zero as $$\vert m\vert \rightarrow \infty $$ and we would like to find an integrable function on $$\mathbf{T}^{1}$$ with $$\widehat{f}(m) \geq a_{m}$$ for all $$m \in \mathbf{Z}$$. Apply Lemma 3.3.2 to the sequence $$\{a_{m} + a_{-m}\}_{m\geq 0}$$ to find a convex sequence $$\{c_{m}\}_{m\geq 0}$$ that dominates $$\{a_{m} + a_{-m}\}_{m\geq 0}$$ and decreases to zero as $$m \rightarrow \infty $$. Extend c m for m < 0 by setting $$c_{m} = c_{\vert m\vert }$$. Now define

(3.3.4)

where F j is the (one-dimensional) Fejér kernel. The convexity of the sequence c m and the positivity of the Fejér kernel imply that f ≥ 0. Lemma 3.3.3 with s = 0 gives that

(3.3.5)

since $$\|F_{j}\|_{L^{1}} = 1$$ for all j. Therefore (3.3.4) defines an integrable function f on $$\mathbf{T}^{1}$$. We now compute the Fourier coefficients of f. Since the series in (3.3.4) converges in L ¹, for $$m \in \mathbf{Z}$$ we have

(3.3.6)

where we used Lemma 3.3.3 with s =  | m | .

Let us now extend this result on $$\mathbf{T}^{n}$$. Let $$(d_{m})_{m\in \mathbf{Z}^{n}}$$ be a positive sequence with d m  → 0 as $$\vert m\vert \rightarrow \infty $$. By Exercise 3.3.2, there exists a positive sequence $$(a_{j})_{j\in \mathbf{Z}}$$ with $$a_{m_{1}}\cdots a_{m_{n}} \geq d_{(m_{1},\ldots,m_{n})}$$ and $$a_{j} \rightarrow 0$$ as $$\vert j\vert \rightarrow \infty $$. Let

where f is the function previously constructed when $$n = 1$$ so that $$\widehat{f}(m) \geq a_{m}$$. It can be seen easily using (3.1.7) that $$\widehat{\mathbf{f}\,}(m) \geq d_{m}$$. □ 

3.3.2 Decay of Fourier Coefficients of Smooth Functions

We next study the decay of the Fourier coefficients of functions that possess a certain amount of smoothness. In this section we see that the decay of the Fourier coefficients reflects the smoothness of the function in a rather precise quantitative way. Conversely, one can infer some information about the smoothness of a function from the decay of its Fourier coefficients.

Definition 3.3.5.

Given $$0 <\gamma < 1$$ and f a function on $$\mathbf{T}^{n}$$, define the homogeneous Lipschitz seminorm of order $$\gamma$$ of f by

and define the homogeneous Lipschitz space of order $$\gamma$$ as

Functions in $$\dot{\varLambda }_{\gamma }(\mathbf{T}^{n})$$ are called homogeneous Lipschitz functions of order $$\gamma$$.

There is an analogous definition for the inhomogeneous norm.

Definition 3.3.6.

For $$0 <\gamma < 1$$ and $$f$$ a function on $$\mathbf{T}^{n}$$, define the inhomogeneous Lipschitz norm of order $$\gamma$$ of f by

Also define the inhomogeneous Lipschitz space of order $$\gamma$$ as

Functions in $$\varLambda _{\gamma }(\mathbf{T}^{n})$$ are called inhomogeneous Lipschitz functions of order $$\gamma$$.

Remark 3.3.7.

Functions in both spaces $$\varLambda _{\gamma }(\mathbf{T}^{n})$$ and $$\dot{\varLambda }_{\gamma }(\mathbf{T}^{n})$$ are obviously continuous and therefore bounded. Moreover, the functional $$\|\,\cdot \,\|_{\varLambda _{\gamma }}$$ is a norm on $$\varLambda _{\gamma }(\mathbf{T}^{n})$$. The positive functional $$\|\,\cdot \,\|_{\dot{\varLambda }_{\gamma }}$$ satisfies the triangle inequality, but it does not satisfy the property $$\|f\|_{\dot{\varLambda }_{\gamma }} = 0\Rightarrow f = 0$$ required to be a norm. It is therefore a seminorm on $$\dot{\varLambda }_{\gamma }(\mathbf{T}^{n})$$. However, if we identify functions whose difference is a constant, we form a space of the equivalence classes $$\dot{\varLambda }_{\gamma }(\mathbf{T}^{n})/\{\mbox{ constants}\}$$ on which $$\|\,\cdot \,\|_{\dot{\varLambda }_{\gamma }}$$ is a norm.

Remark 3.3.8.

We already observed that elements of $$\dot{\varLambda }_{\gamma }(\mathbf{T}^{n})$$ are continuous and thus bounded. Therefore, $$\dot{\varLambda }_{\gamma }(\mathbf{T}^{n}) \subseteq L^{\infty }(\mathbf{T}^{n})$$ in the set-theoretic sense. However, the norm inequality $$\|f\|_{L^{\infty }} \leq C\,\|f\|_{\dot{\varLambda }_{\gamma }}$$ for all $$f \in \dot{\varLambda }_{\gamma }$$ fails for all constants C. For example, take $$f = N +\sin (2\pi x_{1})$$ on $$\mathbf{T}^{n}$$ and let $$N \rightarrow \infty $$ to see that this is the case.

The following theorem indicates how the smoothness of a function is reflected by the decay of its Fourier coefficients.

Theorem 3.3.9.

Let $$s \in \mathbf{Z}$$ with $$s \geq 0$$ .

(a) Suppose that $$\partial ^{\alpha }f$$ exist and are integrable for all $$\vert \alpha \vert \leq s$$ . Then

(3.3.7)

and thus

as $$\vert m\vert \rightarrow \infty $$ . In particular this holds when f lies in $$\mathcal{C}^{s}(\mathbf{T}^{n})$$ .

(b) Suppose that $$\partial ^{\alpha }f$$ exist for all $$\vert \alpha \vert \leq s$$ and whenever $$\vert \alpha \vert = s$$, $$\partial ^{\alpha }f$$ are in $$\dot{\varLambda }_{\gamma }(\mathbf{T}^{n})$$ for some $$0 <\gamma < 1$$ . Then

(3.3.8)

Proof.

Fix $$m \in \mathbf{Z}^{n}\setminus \{0\}$$ and pick a j such that $$\vert m_{j}\vert =\sup _{1\leq k\leq n}\vert m_{k}\vert $$. Then clearly m j ≠ 0. Integrating by parts s times with respect to the variable x j , we obtain

(3.3.9)

where the boundary terms all vanish because of the periodicity of the integrand. Taking absolute values and using $$\vert m\vert \leq \sqrt{n}\,\vert m_{j}\vert,$$ we obtain assertion (3.3.7).

We now turn to the second part of the theorem. Let $$e_{j} = (0,\ldots,1,\ldots,0)$$ be the element of the torus $$\mathbf{T}^{n}$$ whose jth coordinate is one and all the others are zero. A simple change of variables together with the fact that $$e^{\pi i} = -1$$ gives that

which implies that

Now use the estimate

and identity (3.3.9) to conclude the proof of (3.3.8). □ 

The following is an immediate consequence.

Corollary 3.3.10.

Let $$s \in \mathbf{Z}$$ with s ≥ 0.

(a) Suppose that $$\partial ^{\alpha }f$$ exist and are integrable for all $$\vert \alpha \vert \leq s$$ . Then for some constant c n,s we have

(3.3.10)

(b) Suppose that $$\partial ^{\alpha }f$$ exist for all $$\vert \alpha \vert \leq s$$ and whenever $$\vert \alpha \vert = s$$, $$\partial ^{\alpha }f$$ are in $$\dot{\varLambda }_{\gamma }(\mathbf{T}^{n})$$ for some $$0 <\gamma < 1$$ . Then for some constant $$c_{n,s}'$$ we have

(3.3.11)

Remark 3.3.11.

The conclusions of Theorem 3.3.9 and Corollary 3.3.10 are also valid when $$\gamma = 1$$. In this case the spaces $$\dot{\varLambda }_{\gamma }$$ should be replaced by the space $$\mbox{ Lip}\,1$$ equipped with the seminorm

There is a slight lack of uniformity in the notation here, since in the theory of Lipschitz spaces the notation $$\dot{\varLambda }_{1}$$ is usually reserved for the space with seminorm

The following proposition provides a partial converse to Theorem 3.3.9. We denote below by [[s]] the largest integer strictly less than a given real number s. Then [[s]] is equal to the integer part [s] of s, unless s is an integer, in which case [[s]] = [s] − 1.

Proposition 3.3.12.

Let s > 0 and suppose that f is an integrable function on the torus with

(3.3.12)

for all $$m \in \mathbf{Z}^{n}$$ . Then f has partial derivatives of all orders $$\vert \alpha \vert \leq [[s]]$$ , and for $$0 <\gamma < s - [[s]]$$, $$\partial ^{\alpha }f \in \dot{\varLambda }_{\gamma }$$ for all multi-indices $$\alpha$$ satisfying $$\vert \alpha \vert = [[s]]$$ .

Proof.

Since f has an absolutely convergent Fourier series, Proposition 3.2.5 gives that

(3.3.13)

for almost all $$x \in \mathbf{T}^{n}$$.

Suppose that a series $$g =\sum _{m}g_{m}$$ satisfies $$\sum _{m}\|\partial ^{\beta }g_{m}\|_{L^{\infty }} < \infty $$ for all $$\vert \beta \vert \leq M$$. Then the function g is in $$\mathcal{C}^{M}$$ and $$\partial ^{\beta }g =\sum _{m}\partial ^{\beta }g_{m}$$; indeed this can be proved by induction on the degree of the multi-index, since for all $$\vert \beta \vert \leq M - 1$$ we have

where the passage of the limit inside the sum is due to the Lebesgue dominated convergence theorem, which can be applied using the uniform convergence of $$\sum _{m}\partial _{j}\partial ^{\beta }g$$ via the mean value theorem.

Using the preceding observation, the function f in (3.3.13) is $$\mathcal{C}^{[[s]]}(\mathbf{T}^{n})$$ and

for all multi-indices $$(\alpha _{1},\ldots,\alpha _{n})$$ with $$\vert \alpha \vert \leq [[s]]$$, since

which holds because of (3.3.12).

Now suppose that $$\vert \alpha \vert = [[s]]$$ and that $$0 <\gamma < s - [[s]]$$. Then

where we used the relation $$[[s]] +\gamma -s - n < -n$$ to conclude the convergence of the series and the fact that

 □ 

Next we recall the definition of functions of bounded variation.

Definition 3.3.13.

A measurable function f on $$\mathbf{T}^{1}$$ is said to be of bounded variation if it is defined everywhere and

where the supremum is taken over all partitions of the interval [0, 1]. The expression $$\mbox{ Var}(f)$$ is called the total variation of f. The class of functions of bounded variation on $$\mathbf{T}^{1}$$ is denoted by $$BV (\mathbf{T}^{1})$$.

Examples of functions of bounded variation can be constructed as follows: given f ₁, f ₂ nonnegative integrable functions on [0, 1] with

then the periodic function

defined on[0, 1], is of bounded variation. Analogous examples can be constructed when f ₁ and f ₂ are replaced by nonnegative finite Borel measures on [0, 1].

Every function of bounded variation can be represented as the difference of two (not necessarily strictly) increasing functions and thus it has a finite derivative at almost every point. Moreover, for functions of bounded variation, the Lebesgue–Stieltjes integral with respect to df is well defined.

Proposition 3.3.14.

If f is in $$BV (\mathbf{T}^{1})$$ , then

whenever m ≠ 0.

Proof.

Integration by parts gives

where the boundary terms vanish because of periodicity. The conclusion follows from the fact that the norm of the measure df is the total variation of f. □ 

The following chart (Table 3.1) summarizes the decay of Fourier coefficients in terms of scales of spaces measuring the smoothness of the functions. Recall that for $$q \geq 0$$, $$\widehat{f}(m) = o(\vert m\vert ^{-q})$$ means that $$\vert \widehat{f}(m)\vert \,\vert m\vert ^{q} \rightarrow 0$$ as $$\vert m\vert \rightarrow \infty $$ and $$\widehat{f}(m) = O(\vert m\vert ^{-q})$$ means that $$|\widehat{f}(m)|\le C\, |m|^{-q}$$ when | m | is large. In this chart, we denote by $$\mathcal{C}^{s,\gamma }(\mathbf{T}^{n})$$ the space of all $$\mathcal{C}^{s}$$ functions on $$\mathbf{T}^{n}$$, all of whose derivatives of total order s lie in $$\varLambda _{\gamma }(\mathbf{T}^{n})$$, for some $$0 <\gamma < 1$$.

Table 3.1

Interconnection between smoothness of functions and decay of their Fourier coefficients. We take $$0 <\gamma < 1$$ and 1 < p < 2.

SPACE

SEQUENCE OF FOURIER COEFFICIENTS

o(1)

O( | m | ⁻¹)

o( | m | ⁻²)

o( | m | ⁻³)

⋯ 

o( | m | −N ) for all N > 0

3.3.3 Functions with Absolutely Summable Fourier Coefficients

Decay for the Fourier coefficients can also be indirectly deduced from knowledge about the summability of these coefficients. The simplest kind of summability is in the sense of ℓ ¹. It is therefore natural to consider the class of functions on the torus whose Fourier coefficients form an absolutely summable series.

Definition 3.3.15.

An integrable function f on the torus is said to have an absolutely convergent Fourier series if

We denote by $$A(\mathbf{T}^{n})$$ the space of all integrable functions on the torus $$\mathbf{T}^{n}$$ whose Fourier series are absolutely convergent. We then introduce a norm on $$A(\mathbf{T}^{n})$$ by setting

In view of Proposition 3.2.5, every function f in $$A(\mathbf{T}^{n})$$ can be changed on a set of measure zero to be made continuous and under this modification, Fourier inversion

holds for all $$x \in \mathbf{T}^{n}$$. Thus functions in $$A(\mathbf{T}^{n})$$ are continuous and bounded. Moreover, Theorem 3.3.9 yields that every function in $$\mathcal{C}^{n}(\mathbf{T}^{n})$$ whose partial derivatives of order n are in $$\dot{\varLambda }_{\gamma }$$, $$\gamma > 0$$, must lie in $$A(\mathbf{T}^{n})$$. The following theorem gives us a significantly better sufficient condition for a function to be in $$A(\mathbf{T}^{n})$$.

Theorem 3.3.16.

Suppose f is a given function in $$\mathcal{C}^{[n/2]}(\mathbf{T}^{n})$$ and that all partial derivatives of order $$[\frac{n} {2} ]$$ of f lie in $$\dot{\varLambda }_{\gamma }(\mathbf{T}^{n})$$ for some $$\gamma$$ with $$\frac{n} {2} - [\frac{n} {2} ] <\gamma < 1$$ . Then f lies in $$A(\mathbf{T}^{n})$$ and

where $$C(n,\gamma )$$ is a constant depending on n and $$\gamma$$ .

Proof.

For each $$\ell= 0,1,2,\ldots$$, let

We begin by writing

(3.3.14)

where we used the Cauchy-Schwarz inequality and the fact that there are at most $$c_{n}2^{\ell n}$$ points in $$\mathbf{Z}^{n}$$ inside the open ball $$B(0,2^{\ell+1})$$, for some dimensional constant c n .

Notice that for a multi-index $$m = (m_{1},\ldots,m_{n})$$ satisfying $$2^{\ell} \leq \vert m\vert \leq 2^{\ell+1}$$ and for j in $$\{1,\ldots,n\}$$ such that $$\vert m_{j}\vert =\sup _{k}\vert m_{k}\vert $$ we have

(3.3.15)

For 1 ≤ j ≤ n, let e j be the element of $$\mathbf{R}^{n}$$ with zero entries except for the jth coordinate, which is 1, and define

(3.3.16)

Using the elementary fact that $$\vert t\vert \leq \pi \Rightarrow\vert e^{it} - 1\vert \geq 2\vert t\vert /\pi$$, we obtain

(3.3.17)

whenever $$\frac{\vert 2\pi m_{j}\vert } {2^{\ell+2}} \leq \pi$$, which is always true since $$\frac{\vert 2\pi m_{j}\vert } {2^{\ell+2}} \leq \frac{2\pi 2^{\ell+1}} {2^{\ell+2}} \leq \pi$$.

We now have

where we used (3.3.17), (3.3.15), and (3.3.16). We conclude that

which inserted in (3.3.14) yields the desired conclusion since $$\gamma > \frac{n} {2} - [\frac{n} {2} ]$$. □ 

Exercises

3.3.1. Given a sequence $$\{a_{n}\}_{n=0}^{\infty }$$ of positive numbers such that a n  → 0 as $$n \rightarrow \infty $$, find a nonnegative integrable function h on [0, 1] such that

Use this result to deduce a different proof of Lemma 3.3.2.

[Hint: Try $$h = e\sum \limits _{k=0}^{\infty }(\sup \limits _{j\geq k}a_{j} -\sup \limits _{j\geq k+1}a_{j})(k + 2)\chi _{[\frac{k+1} {k+2},1]}$$.]

3.3.2.

Prove that given a positive sequence $$\{d_{m}\}_{m\in \mathbf{Z}^{n}}$$ with d m  → 0 as $$\vert m\vert \rightarrow \infty $$, there exists a positive sequence $$\{a_{j}\}_{j\in \mathbf{Z}}$$ with $$a_{m_{1}}\cdots a_{m_{n}} \geq d_{(m_{1},\ldots,m_{n})}$$ and $$a_{j} \rightarrow 0$$ as $$\vert j\vert \rightarrow \infty $$.

3.3.3.

(a) Use the idea of the proof of Lemma 3.3.3 to prove that if a twice continuously differentiable function $$f \geq 0$$ is defined on $$(0,\infty )$$ and satisfies f′(x) ≤ 0 and f″(x) ≥ 0 for all x > 0, then $$\lim _{x\rightarrow \infty }xf'(x) = 0.$$

(b) Suppose that a continuously differentiable function g is defined on $$(0,\infty )$$ and satisfies g ≥ 0, g′ ≤ 0, and $$\int _{1}^{\infty }g(x)\,dx < +\infty $$. Prove that

3.3.4.

Prove that for $$0 <\gamma <\delta < 1$$ we have $$\|f\|_{\dot{\varLambda }_{\gamma }} \leq C_{n,\gamma,\delta }\|f\|_{\dot{\varLambda }_{\delta }}$$ for all functions f and thus $$\dot{\varLambda }_{\delta }$$ is a subspace of $$\dot{\varLambda }_{\gamma }$$.

3.3.5.

Suppose that f is a differentiable function on $$\mathbf{T}^{1}$$ whose derivative f′ is in $$L^{2}(\mathbf{T}^{1})$$. Prove that $$f \in A(\mathbf{T}^{1})$$ and that

3.3.6.

(a) Prove that the product of two functions in $$A(\mathbf{T}^{n})$$ is also in $$A(\mathbf{T}^{n})$$ and that

(b) Prove that the convolution of two square integrable functions on $$\mathbf{T}^{n}$$ always gives a function in $$A(\mathbf{T}^{n})$$.

3.3.7.

Fix $$0 <\alpha < 1$$ and define f on $$\mathbf{T}^{1}$$ by setting

Prove that the function f lies in $$\dot{\varLambda }_{\alpha }(\mathbf{T}^{1})$$. Conclude that there does not exist positive $$\beta >\alpha$$ such that for all f in $$\dot{\varLambda }_{\alpha }(\mathbf{T}^{1})$$ we have $$\sup _{m\in \mathbf{Z}}\vert m\vert ^{\beta }\vert \widehat{f}(m)\vert < \infty $$.

[Hint: For h ≠ 0 pick $$N \in \mathbf{Z}^{+}$$ such that $$2^{N}\vert h\vert > 1 \geq 2^{N-1}\vert h\vert $$. To estimate the difference $$\vert f(x + h) - f(x)\vert $$, consider the cases k ≤ N and k ≥ N + 1 in the sum.]

3.3.8.

Use without proof that there exists a constant C > 0 such that

to prove that the function

is in $$\dot{\varLambda }_{1/2}(\mathbf{T}^{1})$$ but not in $$A(\mathbf{T}^{1})$$. Conclude that the restriction s > 1∕2 in Theorem 3.3.16 is sharp.

[Hint: Estimate the difference | g(x + h) − g(x) | using the summation by parts identity in Appendix F, taking sums of the sequence $$e^{ik\log k}e^{2\pi ikx}$$ and differences of the sequence $$\frac{e^{2\pi ikh}-1} {k}$$.]

3.3.9. Show that there exist sequences $$\{a_{m}\}_{m\in \mathbf{Z}^{n}}$$ that tend to zero as $$\vert m\vert \rightarrow \infty $$ for which there do not exist functions f in $$L^{1}(\mathbf{T}^{n})$$ with $$\widehat{f}(m) = a_{m}$$ for all m.

[Hint: Suppose the contrary. Then the open mapping theorem would imply the inequality $$\|f\|_{L^{1}(\mathbf{T}^{n})} \leq A\|\widehat{f}\,\|_{\ell^{\infty }(\mathbf{Z}^{n})}$$ for some A > 0. To contradict it, fix a smooth nonzero function h equal to 1 on $$B(0, \frac{1} {4})$$ and supported in $$B(0, \frac{1} {2})$$. For b > 0 define $$g_{b}(x) = h(x)e^{-\pi (1+ib)\vert x\vert ^{2} }$$ and extend g b to a 1-periodic function in each variable on $$\mathbf{R}^{n}$$. Use that $$\widehat{g_{b}}(m) =\int _{\mathbf{R}^{n}}\widehat{h}(y)(1 + ib)^{-n/2}e^{- \frac{\pi }{1+ib}\vert m-y\vert ^{2} }dy$$, and let $$b \rightarrow \infty $$ in the inequality $$\|g_{b}\|_{L^{1}(\mathbf{T}^{n})} \leq A\|\widehat{g_{b}}\,\|_{\ell^{\infty }(\mathbf{Z}^{n})}$$ to obtain a contradiction. ]

3.4 Pointwise Convergence of Fourier Series

In this section we are concerned with the pointwise convergence of the square partial sums and the Fejér means of a function defined on the torus.

3.4.1 Pointwise Convergence of the Fejér Means

We saw in Section 3.1 that the Fejér kernel is an approximate identity. This implies that the Fejér (or Cesàro) means of an $$L^{p}$$ function f on $$\mathbf{T}^{n}$$ converge to it in $$L^{p}$$ for any $$1 \leq p < \infty $$. Moreover, if f is continuous at x ₀, then the means $$(F_{N}^{n} {\ast} f)(x_{0})$$ converge to f(x ₀) as $$N \rightarrow \infty $$ in view of Theorem 1.​2.​19 (2). Although this is a satisfactory result, it is natural to ask what happens for more general functions.

Using properties of the Fejér kernel, we obtain the following one-dimensional result regarding the convergence of the Fejér means:

Theorem 3.4.1.

(Fejér) If a function f in $$L^{1}(\mathbf{T}^{1})$$ has left and right limits at a point x ₀ , denoted by f(x ₀ −) and f(x ₀ +), respectively, then

(3.4.1)

In particular, this is the case for functions of bounded variation.

Proof.

Let us identify $$\mathbf{T}^{1}$$ with [−1∕2, 1∕2]. Given $$\varepsilon > 0$$, find $$\delta \in (0,1/2)$$ such that

(3.4.2)

Using the second expression for F N in (3.1.18), we can find an N ₀ > 0 such that for N ≥ N ₀ we have

(3.4.3)

We now have

Averaging these two identities and using that the integrand is even, we obtain

(3.4.4)

We split the integral in (3.4.4) into two pieces, the integral over $$[0,\delta )$$ and the integral over $$[\delta,1/2]$$. By (3.4.2), the integral over $$[0,\delta )$$ is controlled by $$\varepsilon \int _{\mathbf{T}^{1}}F_{N}(t)\,dt =\varepsilon$$. Also (3.4.3) gives that for $$N \geq N_{0}$$

where c(f, x ₀) is a constant depending on f and x ₀. We have now proved that given $$\varepsilon > 0$$ there exists an N ₀ such that for $$N \geq N_{0}$$ the second expression in (3.4.4) is bounded by $$2\varepsilon \,(c(f,x_{0}) + 1)$$. This proves the required conclusion.

Functions of bounded variation can be written as differences of increasing functions, and since increasing functions have left and right limits everywhere, (3.4.1) holds for these functions. □ 

We continue with an elementary but very useful application of the preceding result.

Proposition 3.4.2.

Let $$x_{0} \in \mathbf{T}^{1}$$ and let f be a complex-valued function on $$\mathbf{T}^{1}$$ . Suppose that the left and right limits of f exist as $$x \rightarrow x_{0}$$ and that the partial sums (Dirichlet means) $$(D_{N} {\ast} f)(x_{0})$$ converge. Then

as $$N \rightarrow \infty $$ .

Proof.

If $$(D_{N} {\ast} f)(x_{0}) \rightarrow L(x_{0})$$ as $$N \rightarrow \infty $$, then

as $$N \rightarrow \infty $$. But $$(F_{N} {\ast} f)(x_{0}) \rightarrow \frac{1} {2}\big(f(x_{0}+) + f(x_{0}-)\big)$$ as $$N \rightarrow \infty $$ in view of Theorem 3.4.1. We conclude that

Thus $$(D_{N} {\ast} f)(x_{0}) \rightarrow \frac{1} {2}\big(f(x_{0}+) + f(x_{0}-)\big)$$ as $$N \rightarrow \infty $$. □ 

This theorem is quite useful when we have a priori knowledge that the Fourier series converges. For instance, consider the following example.

Example 3.4.3.

On (−1∕2, 1∕2) let f(t) = t and f(1∕2) = f(−1∕2) = 1000. Then f is discontinuous at the point $$-1/2 \equiv 1/2$$ but it has left and right limits at this point:

(3.4.5)

Moreover $$\widehat{f}(m) = \frac{i(-1)^{m}} {2\pi m}$$ when m ≠ 0 and $$\widehat{f}(0) = 0$$ by Exercise 3.4.1 (a). It is not hard to see that the series

(3.4.6)

converges for every $$x \in (-1/2,1/2)$$. Indeed, by Appendix F, (3.4.6) equals

which has a limit as $$N \rightarrow \infty $$, since the geometric sums

are bounded above independently of N when x ∈ (−1∕2, 1∕2). We conclude that

whenever | x |  < 1∕2. Moreover, we have that

which is the average of the left and right limits in (3.4.5) as Proposition 3.4.2 states. Exercise 3.4.2 contains other applications of this sort.

3.4.2 Almost Everywhere Convergence of the Fejér Means

We have seen that the Fejér means of a relatively nice function (such as of bounded variation) converge everywhere. What can we say about the Fejér means of a general integrable function? Since the Fejér kernel is an approximate identity that satisfies good estimates, the following result should not come as a surprise.

Theorem 3.4.4.

(a) For $$f \in L^{1}(\mathbf{T}^{n})$$ , let

Then $$\mathcal{H}$$ maps $$L^{1}(\mathbf{T}^{n})$$ to $$L^{1,\infty }(\mathbf{T}^{n})$$ and $$L^{p}(\mathbf{T}^{n})$$ to itself for $$1 < p \leq \infty $$ .

(b) For any function $$f \in L^{1}(\mathbf{T}^{n})$$ , we have as $$N \rightarrow \infty $$

Proof.

It is an elementary fact that $$\vert t\vert \leq \frac{\pi } {2}\Rightarrow\vert \sin t\vert \geq \frac{2} {\pi } \vert t\vert $$; see Appendix E. Using this fact and the expression (3.1.18) we obtain for all t in $$[-\frac{1} {2}, \frac{1} {2}]$$,

For $$t \in \mathbf{R}$$ let us set $$\varphi (t) = (1 + \vert t\vert ^{2})^{-1}$$ and $$\varphi _{\varepsilon }(t) = \frac{1} {\varepsilon } \varphi (\frac{t} {\varepsilon } )$$ for $$\varepsilon > 0$$. For $$x = (x_{1},\ldots,x_{n}) \in \mathbf{R}^{n}$$ and $$\varepsilon > 0$$ we also set

and $$\varPhi _{\varepsilon }(x) =\varepsilon ^{-n}\varPhi (\varepsilon ^{-1}x)$$. Then for $$\vert t\vert \leq \frac{1} {2}$$ we have $$\vert F_{N}(t)\vert \leq \frac{\pi ^{2}} {2}\varphi _{\varepsilon }(t)$$ with $$\varepsilon = (N + 1)^{-1}$$, and for $$y \in [-\frac{1} {2}, \frac{1} {2}]^{n}$$ we have

Now let f be an integrable function on $$\mathbf{T}^{n}$$ and let f ₀ denote its periodic extension on $$\mathbf{R}^{n}$$. For $$x \in [-\frac{1} {2}, \frac{1} {2}]^{n}$$ we have

(3.4.7)

where Q is the cube [−1, 1] n and $$\mathcal{G}$$ is the operator defined on integrable functions on $$\mathbf{R}^{n}$$ by

If we can show that $$\mathcal{G}$$ maps $$L^{1}(\mathbf{R}^{n})$$ to $$L^{1,\infty }(\mathbf{R}^{n})$$, the corresponding conclusion for $$\mathcal{H}$$ on $$\mathbf{T}^{n}$$ would follow from the fact $$\mathcal{H}(f) \leq 5^{n}\mathcal{G}(f_{0}\chi _{Q})$$ proved in (3.4.7) and the sequence of inequalities

Moreover, the $$L^{p}$$ conclusion about $$\mathcal{H}$$ follows from the weak type (1, 1) result and the trivial $$L^{\infty }$$ inequality, in view of the Marcinkiewicz interpolation theorem (Theorem 1.​3.​2). The required weak type (1, 1) estimate for $$\mathcal{G}$$ on $$\mathbf{R}^{n}$$ is a consequence of Lemma 3.4.5. Modulo the proof of this lemma, part (a) of the theorem is proved.

To prove the statement in part (b) observe that for $$f \in \mathcal{C}^{\infty }(\mathbf{T}^{n})$$, which is a dense subspace of L ¹, we have $$F_{N}^{n} {\ast} f \rightarrow f$$ uniformly on $$\mathbf{T}^{n}$$ as $$N \rightarrow \infty $$, since the sequence $$\{F_{N}\}_{N}$$ is an approximate identity. Since by part (a), $$\mathcal{H}$$ maps $$L^{1}(\mathbf{T}^{n})$$ to $$L^{1,\infty }(\mathbf{T}^{n})$$, Theorem 2.​1.​14 yields that for $$f \in L^{1}(\mathbf{T}^{n})$$, $$F_{N}^{n} {\ast} f \rightarrow f$$ a.e. □ 

We now prove the weak type (1, 1) boundedness of $$\mathcal{G}$$ used earlier.

Lemma 3.4.5.

Let $$\varPhi (x_{1},\ldots,x_{n}) = (1 + \vert x_{1}\vert ^{2})^{-1}\cdots (1 + \vert x_{n}\vert ^{2})^{-1}$$ and for $$\varepsilon > 0$$ let $$\varPhi _{\varepsilon }(x) =\varepsilon ^{-n}\varPhi (\varepsilon ^{-1}x)$$ . Then the maximal operator

maps $$L^{1}(\mathbf{R}^{n})$$ to $$L^{1,\infty }(\mathbf{R}^{n})$$ .

Proof.

Let I ₀ = [−1, 1] and $$I_{k} =\{ t \in \mathbf{R}:\, 2^{k-1} \leq \vert t\vert \leq 2^{k}\}$$ for $$k = 1,2,\ldots$$. Also, let $$\widetilde{I}_{k}$$ be the convex hull of I k , that is, the interval [−2 k , 2 k ]. For $$a_{2},\ldots,a_{n}$$ fixed positive numbers, let $$M_{a_{2},\ldots,a_{n}}$$ be the maximal operator obtained by averaging a function on $$\mathbf{R}^{n}$$ over all products of closed intervals $$J_{1} \times \ldots \times J_{n}$$ containing a given point with

In view of Exercise 2.1.9(c), we have that $$M_{a_{2},\ldots,a_{n}}$$ maps L ¹ to $$L^{1,\infty }$$ with some constant independent of the a j ’s. (This is due to the nice doubling property of this family of rectangles.) For a fixed $$\varepsilon > 0$$ we estimate the expression

Split $$\mathbf{R}^{n}$$ into n! regions of the form $$\vert y_{j_{1}}\vert \geq \ldots \geq \vert y_{j_{n}}\vert,$$ where $$\{j_{1},\ldots,j_{n}\}$$ is a permutation of the set $$\{1,\ldots,n\}$$ and $$y = (y_{1},\ldots,y_{n})$$. By symmetry, we examine the region $$\mathcal{R}$$ where $$\vert y_{1}\vert \geq \ldots \geq \vert y_{n}\vert $$. Then for some constant C > 0 we have

and the last expression is trivially controlled by the corresponding expression, where the I k ’s are replaced by the $$\widetilde{I}_{k}$$’s. This, in turn, is controlled by

(3.4.8)

Now set $$s_{2} = k_{1} - k_{2},\ldots,s_{n} = k_{1} - k_{n}$$, observe that s j  ≥ 0, use that

and change the indices of summation to estimate the expression in (3.4.8) by

Argue similarly for the remaining regions $$\vert y_{j_{1}}\vert \geq \ldots \geq \vert y_{j_{n}}\vert $$. Finally, translate to an arbitrary point x to obtain the estimate

Now take the supremum over all $$\varepsilon > 0$$ and use the fact that the maximal functions $$M_{s_{2},\ldots,s_{n}}$$ map L ¹ to $$L^{1,\infty }$$ uniformly in $$s_{2},\ldots,s_{n}$$ as well as the result of Exercise 1.4.10 to obtain the desired conclusion for $$\mathcal{G}$$. □ 

3.4.3 Pointwise Divergence of the Dirichlet Means

We now pass to the more difficult question of convergence of the square partial sums of a Fourier series. It is natural to start our investigation with the class of continuous functions. Do the partial sums of the Fourier series of continuous functions converge pointwise? The following simple proposition warns about the behavior of partial sums.

Proposition 3.4.6.

(a) (duBois Reymond) There exists a continuous function f on $$\mathbf{T}^{1}$$ whose partial sums diverge at a point. Precisely, for some point $$x_{0} \in \mathbf{T}^{1}$$ we have

(b) There exists a continuous function F on $$\mathbf{T}^{n}$$ and $$x_{0} \in \mathbf{T}^{1}$$ such that the sequence

for all $$x_{2},\ldots,x_{m}$$ in $$\mathbf{T}^{1}$$ .

Proof.

The proof of part (b) is obtained by considering the continuous function $$F(x_{1},\ldots,x_{n}) = f(x_{1})$$, where f is as in part (a). Then we have

and thus the square partial sums of F diverge on the (n − 1)-dimensional plane $$\{(x_{0},x_{2},\ldots,x_{n}):\, x_{2},\ldots,x_{n} \in \mathbf{T}^{1}\}$$.

We now prove part (a) using functional analysis. For a constructive proof, see Exercise 3.4.7. Let $$C(\mathbf{T}^{1})$$ be the Banach space of all continuous functions on the circle equipped with the $$L^{\infty }$$ norm. Consider the continuous linear functionals

on $$C(\mathbf{T}^{1})$$ for $$N = 1,2,\ldots$$. We show that the norms of the T N ’s on $$C(\mathbf{T}^{1})$$ converge to infinity as $$N \rightarrow \infty $$. To see this, given any integer N ≥ 100, let $$\varphi _{N}(x)$$ be a continuous even function on $$[-\frac{1} {2}, \frac{1} {2}]$$ that is bounded by 1 and is equal to the sign of D N (x) except at small intervals of length (2N + 1)⁻² around the 2N + 1 zeros of D N . Call the union of all these intervals B N and set $$A_{N} = [-\frac{1} {2}, \frac{1} {2}]\setminus B_{N}$$. Then

Using this estimate we obtain

It follows that the norms of the linear functionals T N are not uniformly bounded. The uniform boundedness principle now implies the existence of a function $$f \in C(\mathbf{T}^{1})$$ and of a sequence $$N_{j} \rightarrow \infty $$ such that

as $$j \rightarrow \infty $$. The Fourier series of this f diverges at x ₀ = 0. □ 

3.4.4 Pointwise Convergence of the Dirichlet Means

We have seen that continuous functions may have divergent Fourier series. How about Lipschitz continuous functions? As it turns out, there is a more general condition that implies convergence for the Fourier series of functions that satisfy a certain integrability condition.

Theorem 3.4.7.

(Dini) Let f be an integrable function on $$\mathbf{T}^{1}$$ , let t ₀ be a point on $$\mathbf{T}^{1}$$ for which f(t ₀ ) is defined and assume that

(3.4.9)

Then $$(D_{N}^{n} {\ast} f)(t_{0}) \rightarrow f(t_{0})$$ as $$N \rightarrow \infty $$ .

(Tonelli) Let f be an integrable function on $$\mathbf{T}^{n}$$ and let $$a = (a_{1},\ldots,a_{n}) \in \mathbf{T}^{n}$$ . If f is defined at a and

(3.4.10)

then we have $$(D_{N}^{n} {\ast} f)(a) \rightarrow f(a)$$ as $$N \rightarrow \infty $$ .

Proof.

Since the one-dimensional result is contained in the multidimensional one, we prove the latter. Replacing f(x) by f(x + a) − f(a), we may assume that a = 0 and f(a) = 0. Using identities (3.1.15) and (3.1.14), we can write

(3.4.11)

Expand out the product to express the integrand as a sum of terms of the form

(3.4.12)

where I is a subset of $$\{1,2,\ldots,n\}$$; here we use the convention that the product over an empty set of indices is 1. The function f I inside the curly brackets in (3.4.12) is integrable on $$[-\frac{1} {2}, \frac{1} {2})^{n}$$ except possibly in a neighborhood of the origin, since $$\vert \sin (\pi x_{j})\vert \geq 2\vert x_{j}\vert $$ when $$\vert x_{j}\vert \leq \frac{1} {2}$$. But condition (3.4.10) with a = 0 and f(a) = 0 guarantees that f I is also integrable in a neighborhood of the origin. Expressing the sines and cosines in (3.4.12) in terms of exponentials, we obtain that the integral of (3.4.12) over $$[-\frac{1} {2}, \frac{1} {2})^{n}$$ is a finite linear combination of Fourier coefficients of f I at the points $$(\pm N,\ldots,\pm N) \in \mathbf{Z}^{n}$$. Applying Lemma 3.3.1 yields that the expression in (3.4.11) tends to zero as $$N \rightarrow \infty $$. □ 

The following are consequences of this test.

Corollary 3.4.8.

(a) (Riemann’s principle of localization) Let f be an integrable function on $$\mathbf{T}^{1}$$ that vanishes on an open interval I. Then D N ∗ f converges to zero on the interval I.

(b) Let $$a = (a_{1},\ldots,a_{n}) \in \mathbf{T}^{n}$$ and suppose that an integrable function f on $$\mathbf{T}^{n}$$ is constant on the cross

where $$0 <\delta _{j} < 1/2$$ are fixed. Then $$(D_{N}^{n} {\ast} f)(a) \rightarrow f(a)$$ as $$N \rightarrow \infty $$ .

Proof.

(a) Let t ₀ ∈ I. If f vanishes on I, condition (3.4.9) holds, since the function $$t\mapsto f(t + t_{0}) - f(t_{0})$$ vanishes on − t ₀ + I, which is an interval containing the origin, and is integrable outside − t ₀ + I. Thus $$(D_{N} {\ast} f)(t_{0}) \rightarrow f(t_{0}) = 0$$ for every t ₀ ∈ I.

(b) We need to show that the function

is integrable over $$\mathbf{T}^{n} = [-1/2,1/2)^{n}$$. The integral of this function over $$\mathbf{T}^{n}$$ is equal to its integral over the region

since f(x + a) − f(a) vanishes whenever $$\vert x_{j}\vert <\delta _{j}$$ for some $$j \in \{ 1,2,\ldots,n\}$$. But on S we have that

and this function is integrable over S, since f is. We deduce that (3.4.10) holds. □ 

Corollary 3.4.9.

Let $$a \in \mathbf{T}^{n}$$ and suppose that $$f \in L^{1}(\mathbf{T}^{n})$$ satisfies

for some $$C,\varepsilon _{j} > 0$$ and for all $$x \in \mathbf{T}^{n}$$ . Then the square partial sums $$(D_{N}^{n} {\ast} f)(a)$$ converge to f(a).

Proof.

Note that condition (3.4.10) holds. □ 

Corollary 3.4.10.

(Dirichlet) If f is defined on $$\mathbf{T}^{1}$$ and is a differentiable function at a point a in $$\mathbf{T}^{1}$$ , then $$(D_{N} {\ast} f)(a) \rightarrow f(a)$$ .

Proof.

There exists a $$\delta > 0$$ (say less than 1∕2) such that | f(x) − f(a) | ∕ | x − a | is bounded by | f′(a) | + 1 for $$\vert x - a\vert \leq \delta$$. Also $$\vert f(x) - f(a)\vert /\vert x - a\vert $$ is bounded by $$\vert f(x) - f(a)\vert /\delta$$ when $$\vert x - a\vert >\delta$$. It follows that condition (3.4.9) holds. □ 

Exercises

3.4.1. Identify $$\mathbf{T}^{1}$$ with [−1∕2, 1∕2) and fix 0 < b < 1∕2. Prove the following:

(a)

The mth Fourier coefficient of the function x is $$i\frac{(-1)^{m}} {2\pi m}$$ when m ≠ 0 and 0 when m = 0.

(b)

The mth Fourier coefficient of the function χ [−b, b] is $$\frac{\sin (2\pi bm)} {m\pi }$$ when m ≠ 0 and 2b when m = 0.

(c)

The mth Fourier coefficient of the function $$\big(1 -\frac{\vert x\vert } {b} \big)_{\!+}$$ is $$\,\frac{\sin ^{2}(\pi bm)} {bm^{2}\pi ^{2}}$$ when m ≠ 0 and b when m = 0.

(d)

The mth Fourier coefficient of the function | x | is $$- \frac{1} {2m^{2}\pi ^{2}} + \frac{(-1)^{m}} {2m^{2}\pi ^{2}}$$ when m ≠ 0 and $$\frac{1} {4}$$ when m = 0.

(e)

The mth Fourier coefficient of the function x ² is $$\frac{(-1)^{m}} {2m^{2}\pi ^{2}}$$ when m ≠ 0 and $$\frac{1} {12}$$ when m = 0.

(f)

The mth Fourier coefficient of the function $$\cosh (2\pi x)$$ is $$\frac{(-1)^{m}} {1+m^{2}} \, \frac{\sinh \pi } {\pi }.$$

(g)

The mth Fourier coefficient of the function $$\sinh (2\pi x)$$ is $$\frac{im(-1)^{m}} {1+m^{2}} \, \frac{\sinh \pi } {\pi }.$$

3.4.2. Use Exercise 3.4.1 and Proposition 3.4.2 to prove that

3.4.3. Let M > N be given positive integers.

(a) For $$f \in L^{1}(\mathbf{T}^{1})$$, prove the following identity:

(b) (G. H. Hardy) Suppose that a function f on $$\mathbf{T}^{1}$$ satisfies the following condition: for any $$\varepsilon > 0$$ there exists an a > 1 and a k ₀ > 0 such that for all $$k \geq k_{0}$$ we have

Use part (a) to prove that if $$(F_{N} {\ast} f)(x)$$ converges (uniformly) to A(x) as $$N \rightarrow \infty $$, then (D N ∗ f)(x) also converges (uniformly) to A(x) as $$N \rightarrow \infty $$.

3.4.4. Use Proposition 3.4.2 to show that for $$0 < b < \frac{1} {2}$$ we have

[Hint: Use Exercise 3.4.1(b).]

3.4.5. Let f be an integrable function on $$\mathbf{T}^{n}$$ and g be a bounded function on $$\mathbf{T}^{n}$$ and let K be a compact subset of $$\mathbf{T}^{n}$$. Consider the family $$\mathcal{F} =\{ f_{w}:\,\, w \in K\}$$, where $$f_{w}(x) = f(x - w)g(x)$$ for all $$x \in \mathbf{T}^{n}$$. Prove that the Riemann–Lebesgue lemma holds uniformly for the family $$\mathcal{F}$$. This means that given $$\varepsilon > 0$$ there exists an N ₀(K) > 0 such that for $$\vert m\vert \geq N_{0}$$ we have $$\vert \widehat{f_{w}}(m)\vert \leq \varepsilon$$ for all w ∈ K.

3.4.6. Prove the following version of Corollary 3.4.8 (b). Suppose that a function f on $$\mathbf{T}^{n}$$ is constant on the cross $$U =\{ (x_{1},\ldots,x_{n}) \in \mathbf{T}^{n}:\,\, \vert x_{j} - a_{j}\vert <\delta \,\, \mbox{ for some $j$}\},$$ for some $$\delta < 1/2$$. Then D N n ∗ f converges to f(a) uniformly on compact subsets of the box $$W =\{ (x_{1},\ldots,x_{n}) \in \mathbf{T}^{n}:\,\, \vert x_{j}\vert <\delta \,\, \mbox{ for all $j$}\}$$.

[Hint: Use Exercise 3.4.5.]

3.4.7. Follow the steps given to obtain a constructive proof of the existence of a continuous function whose Fourier series diverges at a point. Identify $$\mathbf{T}^{1}$$ with [0, 1) and define

(a) Prove that $$\widehat{g}(m) = 1/m$$ when m ≠ 0 and zero otherwise.

(b) Prove that for all nonnegative integers M and N we have

when $$M \geq 2N$$ and

when M < 2N. Conclude that there exists a constant C > 0 such that for all M, N, and x ≠ 0 we have

(c) Show that there exists a constant C ₁ > 0 such that

(d) Let $$\lambda _{k} = 1 + e^{e^{k} }$$. Define

and prove that f is continuous on $$\mathbf{T}^{1}$$ and that its Fourier series converges at every x ≠ 0, but $$\limsup _{M\rightarrow \infty }\vert (f {\ast} D_{M})(0)\vert = \infty $$.

[Hint: Take $$M = e^{e^{m} }$$ with $$m \rightarrow \infty $$. The inequality in part (b) follows by summation by parts.]

3.5 A Tauberian theorem and Functions of Bounded Variation

The relation between the partial sums of a Fourier series and the Fejér means is a particular situation of a relation between sequences of complex numbers and their arithmetic means. Given a sequence $$\{a_{k}\}_{k=0}^{\infty }$$ of complex numbers, we denote its partial sums by

for $$N \geq 0$$, and its arithmetic or Cesàro means by

A classical result says that if $$s_{N} \rightarrow L$$ as $$N \rightarrow \infty $$, then $$\sigma _{N} \rightarrow L$$ as $$N \rightarrow \infty $$. The converse is not true, as the example $$a_{k} = (-1)^{k}$$ indicates. But in a particular situation the reverse implication holds.

3.5.1 A Tauberian theorem

We have the following result concerning the convergence of $$\{s_{k}\}_{k=0}^{\infty }$$ as a consequence of that of $$\{\sigma _{k}\}_{k=0}^{\infty }$$.

Theorem 3.5.1.

(a) Suppose that for a sequence $$\{a_{k}\}_{k=0}^{\infty }$$ of complex numbers we have that $$\sigma _{N} \rightarrow L$$ as $$N \rightarrow \infty $$ and that $$\vert ka_{k}\vert \leq M < \infty $$ for all $$k = 0,1,2,\ldots$$ . Then $$s_{k} \rightarrow L$$ as $$k \rightarrow \infty $$ .

(b) Let X be a nonempty set. Suppose that for a sequence $$\{a_{k}(x)\}_{k=0}^{\infty }$$ of complex-valued functions on X we have that $$\sigma _{N}(x) \rightarrow L(x)$$ uniformly in x ∈ X as $$N \rightarrow \infty $$ and that $$\sup _{k\geq 0}\sup _{x\in X}\vert ka_{k}(x)\vert \leq M < \infty $$ . Then $$s_{k}(x) \rightarrow L(x)$$ uniformly in x ∈ X as $$k \rightarrow \infty $$ .

Proof.

We prove part (b), noting that the proof of part (a) is subsumed in that of (b).

For $$0 \leq k < m < \infty $$ we have

Therefore we have

and thus

(3.5.1)

Notice that

(3.5.2)

Now fix $$\varepsilon > 0$$ such that $$\varepsilon < 1$$. For each $$k \in \mathbf{Z}^{+}$$ pick an $$m_{k} \in \{ k,k + 1,\ldots,2k\}$$ such that $$\frac{m_{k}} {k} \rightarrow 1+\varepsilon$$. Then $$\frac{m_{k}+1} {m_{k}-k}$$ converges to $$\varepsilon ^{-1} + 1$$ as $$k \rightarrow \infty $$, hence it is bounded by some constant $$C_{\varepsilon }$$. Then (3.5.1) and (3.5.2) with m k in place of m yield

Taking the $$\limsup _{k\rightarrow \infty }$$ in the preceding inequality and using that

which is a consequence of the hypothesis that $$\sigma _{k}$$ (and thus $$\sigma _{m_{k}}$$) converges to L uniformly, we obtain

In view of the Taylor expansion

which is valid for $$0 <\varepsilon < 1$$, we conclude that

for some absolute constant c > 0. Since $$\varepsilon > 0$$ was arbitrary, we finally deduce that s k (x) converges uniformly to the same limit as $$\sigma _{k}(x)$$, which is L(x). □ 

Corollary 3.5.2.

Suppose that a function f on $$\mathbf{T}^{1}$$ is continuous and there is a constant M > 0 such that $$\vert \widehat{f}(m)\vert \leq M\vert m\vert ^{-1}$$ for all $$m \in \mathbf{Z}^{+}\setminus \{0\}$$ . Then the Fourier series of f converges uniformly to f. In particular, if f is a continuous function of bounded variation on the circle, then $$f {\ast} D_{N} \rightarrow f$$ uniformly on $$\mathbf{T}^{1}$$ as $$N \rightarrow \infty $$ .

Proof.

The Fejér means $$\{F_{N}\}_{N=0}^{\infty }$$ are an approximate identity on $$\mathbf{T}^{n}$$ (Proposition 3.1.10) and so F N ∗ f converge uniformly to f on $$\mathbf{T}^{1}$$ as $$N \rightarrow \infty $$ in view of Theorem 1.​2.​19 (2). Moreover, we have $$\vert m\vert \,\vert \widehat{f}(m)\vert \leq M$$ for all $$m \in \mathbf{Z}$$. It follows from Theorem 3.5.1 that D N ∗ f converges uniformly to f.

If, additionally, f is a function of bounded variation, then $$\vert m\vert \,\vert \widehat{f}(m)\vert \leq \frac{1} {2\pi }\mbox{ Var}(f)$$, as shown in Proposition 3.3.14. Then the claimed conclusion follows. □ 

3.5.2 The sine integral function

We examine a few useful properties of the antiderivative of sin(t)∕t.

Definition 3.5.3.

For $$0 \leq x < \infty $$ define the sine integral function

(3.5.3)

Integrating by parts we write

from which it follows that the limit of Si(x) as $$x \rightarrow \infty $$ exists and is equal to

To precisely evaluate the limit of Si(x) as $$x \rightarrow \infty $$ we write

which converges to π∕2 + 0 as $$N \rightarrow \infty $$, in view of the Riemann-Lebesgue lemma (Proposition 3.3.1), since the function inside the curly brackets is integrable over the circle. We conclude that $$\lim _{x\rightarrow \infty }Si(x) =\pi /2$$.

Note that Si′ vanishes at $$n\pi$$, $$n = 0,1,2,\ldots$$ and $$Si''(n\pi ) = (-1)^{n}/n\pi$$. Consequently, Si(x) has local maxima at the points $$\pi$$, $$3\pi$$, $$5\pi$$, $$\ldots$$ and local minima at the points 2π, 4π, 6π, $$\ldots$$. Moreover, it is increasing on the intervals $$[2k\pi,(2k + 1)\pi ]$$ and decreasing on $$[(2k + 1)\pi,(2k + 2)\pi ]$$, $$k = 0,1,2,\ldots$$. Also, observe that

and likewise we can prove the remaining inequalities in the sequence

Similarly, one can show that

Hence Si(π) is the absolute maximum of Si(x) on [0, ∞), while 0 is the absolute minimum of Si(x) on [0, ∞); Si(π) is the absolute minimum of Si(x) on [π, ∞).

3.5.3 Further properties of functions of bounded variation

Next we have the following theorem concerning functions of bounded variation. Recall that functions of bounded variation are differences of increasing functions and thus have left and right limits at every point.

Theorem 3.5.4.

Let $$0 <\delta \leq 1/2$$ . Suppose that f is an integrable function on $$\mathbf{T}^{1}$$ which is of bounded variation on the neighborhood $$[t_{0}-\delta,t_{0}+\delta ]$$ of the point $$t_{0} \in \mathbf{T}^{1}$$ . Then

Proof.

We write W for the neighborhood $$(-\delta,\delta )$$ of 0, $$F_{t_{0}}(t) = \frac{f(t_{0}-t)+f(t_{0}+t)} {2}$$, and $$L_{t_{0}} = \frac{f(t_{0}+)+f(t_{0}-)} {2}$$. We have

hence, averaging yields

Therefore we have

and since in the second integral $$\vert t\vert \geq \delta$$, the Riemann-Lebesgue lemma shows that the second term is o(1), i.e., it tends to zero as $$N \rightarrow \infty $$. We now show that the first integral also goes to zero. We write

(3.5.4)

but since the function $$\frac{1} {\pi t} - \frac{1} {\sin (\pi t)}$$ remains bounded on $$[-\frac{1} {2}, \frac{1} {2}]$$ (Exercise 3.1.5 (a)), it follows from the Riemann-Lebesgue lemma that the second term is o(1) as $$N \rightarrow \infty $$. Consequently,

as $$N \rightarrow \infty $$. To prove the required conclusion, it will suffice to show that

(3.5.5)

as $$N \rightarrow \infty $$. Let Si(t) be as defined in (3.5.3). We express the integral in (3.5.5) as

(3.5.6)

Integrating by parts we obtain that (3.5.6) is equal to

(3.5.7)

Letting $$N \rightarrow \infty $$ and using the Lebesgue dominated convergence theorem, we conclude that (3.5.7) converges to

noticing that $$L_{t_{0}} = F_{t_{0}}(0+)$$. □ 

Next, we obtain an explicit bound for the partial sums of functions of bounded variation. Let Si(t) be as in (3.5.3).

Theorem 3.5.5.

Suppose that f is a function of bounded variation on the circle $$\mathbf{T}^{1}$$ . Then the partial sums of the Fourier series of f are uniformly bounded, in particular, we have

(3.5.8)

Proof.

We take $$\delta = 1/2$$ in the proof of the preceding theorem. For a point $$t_{0} \in \mathbf{T}^{1}$$, let $$F_{t_{0}}(t) = \frac{f(t_{0}-t)+f(t_{0}+t)} {2}$$. We have that

(3.5.9)

Using that $$\big\vert \frac{1} {\sin (\pi t)} -\frac{1} {\pi t} \big\vert \leq 1 -\frac{2} {\pi }$$ when $$\vert t\vert \leq \frac{1} {2}$$ (Exercise 3.1.5 (a)), we obtain that the second integral in (3.5.9) is bounded by $$(1 -\frac{2} {\pi } )\|f\|_{L^{\infty }}$$. Integrating by parts as in the proof of the preceding theorem, we express the first integral in (3.5.9) as

(3.5.10)

which is bounded (in absolute value) by $$\|f\|_{L^{\infty }}Si(\pi ) + Si(\pi )\mbox{ Var}(f)$$. Assertion (3.5.8) now follows. □ 

3.5.4 Gibbs phenomenon

It is not reasonable to expect that the Fourier series of a discontinuous function converges uniformly in a neighborhood of a discontinuity. The lack of uniformity in the convergence can be measured in terms of the worst jump, called the overshoot. The exact form of nonuniform convergence is illustrated in the following example:

Example 3.5.6.

Consider the function

(3.5.11)

Clearly h(t) is a function of bounded variation and is continuous except at the point t = 0 at which it has a jump discontinuity. Since h is an odd function, its Fourier coefficients are

when m ≠ 0 and $$\widehat{h}(0) = 0$$. The partial sums of the Fourier series of h are

Notice that

Then, if we define $$d(s) = \frac{1} {\sin (\pi s)} -\frac{1} {\pi s}$$, we can write

Notice that d(s) is continuous at zero and d(0) = 0, while $$\lim _{s\rightarrow 0}\frac{d(s)} {s} = \frac{\pi } {6}$$; thus d is a differentiable function on $$[0, \frac{1} {2}]$$ and $$d'(0) = \frac{\pi } {6}$$. Moreover, $$\lim _{s\rightarrow 0}d'(s) = d'(0)$$, thus d is continuously differentiable on $$[0, \frac{1} {2}]$$. Additionally both d and d′ are nonnegative and increasing on $$[0, \frac{1} {2}]$$, thus $$d' \leq \frac{4} {\pi }$$; see the hint of Exercise 3.1.5. It follows that

and the preceding expression is bounded in absolute value by $$(\frac{d(\frac{1} {2} )} {\pi } + \frac{1} {2} \frac{d'(\frac{1} {2} )} {\pi } ) \frac{1} {2N+1}$$.

We deduce that

where $$O\big( \frac{1} {2N+1}\big)$$ is a function bounded by $$\frac{1} {\pi } \frac{1} {2N+1}$$. Consequently,

Hence for $$t \in (0,-\frac{1} {2}]$$ we have $$\lim _{N\rightarrow \infty }(h {\ast} D_{N})(t) = -t + \frac{1} {\pi } \frac{\pi } {2} = \frac{1} {2} - t$$ as expected. Analogously for $$t \in [-\frac{1} {2},0)$$ we have $$\lim _{N\rightarrow \infty }(h {\ast} D_{N})(t) = -\frac{1} {2} - t$$. Also for t = 0, $$\lim _{N\rightarrow \infty }(h {\ast} D_{N})(0) = 0$$. Thus the Fourier series of h at zero converges to the “fair” value of the average of h(0+) and h(0−) which happens to be h(0) = 0.

To quantitatively estimate the nonuniformity of the convergence of (h ∗ D N )(t) we note that

Thus for all $$N = 1,2,\ldots$$ and $$t \in (0, \frac{1} {2}]$$ we have

Also, for any sequence $$t_{N} \rightarrow 0+$$ we have

(3.5.12)

while if for each N we consider the value t N  = 1∕(2N + 1), we obtain that

(3.5.13)

Fig. 3.3

The partial sums (h ∗ D ₄₀)(t) showing the overshoot of approximately 9% of the jump of h at zero.

The quantity $$.08949\ldots$$ is called the overshoot of the partial sums of the Fourier series of the function h in a neighborhood of zero. See Figure 3.3.

We now examine the preceding phenomenon in the setting of functions of bounded variation. These functions can be written as a differences of two increasing functions, so they have countable sets of discontinuities. Suppose we are given a function $$f \in L^{1}(\mathbf{T}^{1})$$ of bounded variation, and for the sake of simplicity, let us consider the situation where it has exactly one discontinuity, say at the point $$t_{0} \in \mathbf{T}^{1}$$. Consider the function h defined in (3.5.11) and define

(3.5.14)

Now the function f − f ₀ is of bounded variation and is also continuous and satisfies $$(f - f_{0})(t_{0}) = 0$$. In view of Corollary 3.5.2, the Fourier series of f − f ₀ converges uniformly to f − f ₀ and so the lack of uniformity of the convergence of the partial sums of f is due to the presence of f ₀.

We express these observations as a theorem.

Theorem 3.5.7.

(a) Let h be defined in (3.5.11) . Then the set of accumulation points of sets of the form $$\{(h {\ast} D_{N})(t_{N})\}_{N\in \mathbf{Z}^{+}}$$ , where t N ∈ [0,1∕2], is the interval

In particular if $$t_{N} \rightarrow 0$$ such that $$Nt_{N} \rightarrow \frac{1} {2}$$ , then

(b) Let f be a function of bounded variation on the circle with a single discontinuity at the point t ₀ , such that $$f(t_{0}+) - f(t_{0}-) > 0$$ . Then the set of accumulation points of sets of the form $$\{(f {\ast} D_{N})(t_{N})\}_{N\in \mathbf{Z}^{+}}$$ , where $$t_{N} \in [t_{0},t_{0}+\delta ]$$ , for some $$\delta > 0$$ , is the interval

In particular if $$t_{N} \rightarrow t_{0}+$$ such that $$N(t_{N} - t_{0}) \rightarrow \frac{1} {2}$$ , then

Proof.

(a) Since $$h \geq 0$$ on $$(0, \frac{1} {2}]$$ and $$(h {\ast} D_{N})(t) \rightarrow \frac{1} {2} - t$$ for $$0 < t \leq \frac{1} {2}$$ we have that all accumulation points of sequences $$(h {\ast} D_{N})(t_{N})$$ are nonnegative. We showed in (3.5.12) that all accumulation points of sequences $$(h {\ast} D_{N})(t_{N}) - h(t_{N})$$ are at most $$\frac{Si(\pi )} {\pi } -\frac{1} {2}$$; but $$h(t_{N}) \leq \frac{1} {2}$$, when $$t_{N} \in [0, \frac{1} {2}]$$, hence all accumulation points of sequences $$(h {\ast} D_{N})(t_{N})$$ are at most $$\frac{Si(\pi )} {\pi }$$ and thus contained in $$\big[0, \frac{Si(\pi )} {\pi } \big]$$. Also, 0 is attained as the accumulation point of $$(h {\ast} D_{N})(0)$$ and the number $$\frac{Si(\pi )} {\pi }$$ is attained as the accumulation point of the sequence $$(h {\ast} D_{N})( \frac{1} {2N+1})$$ as shown in (3.5.13); notice that the same assertion is valid for any other sequence $$t_{N} \rightarrow 0$$ such that $$Nt_{N} \rightarrow \frac{1} {2}$$. Now, since the functions $$(h {\ast} D_{N})(t)$$ are continuous, given any c in $$\big[0, \tfrac{Si(\pi )} {\pi } \big]$$, there is a t N ′ between 0 and $$\frac{1} {2N+1}$$ such that $$(h {\ast} D_{N})(t_{N}') = c$$ for all N; this shows that the set of accumulation points of sequences of the form $$\{(f {\ast} D_{N})(t_{N})\}_{N\in \mathbf{Z}^{+}}$$ is the interval $$\big[0, \tfrac{Si(\pi )} {\pi } \big]$$.

(b) To examine the behavior of f ∗ D N near the point of a single jump discontinuity t ₀ of f, we reduce matters to the preceding situation as alluded earlier, by introducing the function f ₀ defined in (3.5.14). Then for a sequence t N converging to t ₀ from the right we have

Applying $$\limsup _{N\rightarrow \infty }$$ and using (3.5.12), we obtain as $$N \rightarrow \infty $$

where we used the following consequence of Theorem 3.5.4

This shows that all accumulation points of sequences of the form $$\{(f {\ast} D_{N})(t_{N})\}_{N\in \mathbf{Z}^{+}}$$ are at most $$\big(f(t_{0}+) - f(t_{0}-)\big)\tfrac{Si(\pi )} {\pi } + \tfrac{1} {2}\big(f(t_{0}+) + f(t_{0}-)\big)$$. Also, since all accumulation points of sequences $$(h {\ast} D_{N})(t_{N})$$ are nonnegative, when t N lies to the right of t ₀, it follows from the identity $$f = (f - f_{0}) + f_{0}$$ and (3.5.14) that all accumulation points of $$\{(f {\ast} D_{N})(t_{N})\}_{N\in \mathbf{Z}^{+}}$$ are at least $$\tfrac{1} {2}\big(f(t_{0}+) + f(t_{0}-)\big)$$. As in part (a) the intermediate value theorem implies that every point in the interval having the aforementioned endpoints is also an accumulation point for the sequence at hand. □ 

Exercises

3.5.1. Let Si(t) be the sine function as defined in (3.5.3).

(a) Prove that $$\vert \frac{\pi }{2} - Si(t)\vert \leq \frac{2} {t}$$.

(b) Show that $$Si(Nt) \rightarrow \tfrac{\pi } {2}$$ uniformly in $$t \in [\delta,\infty )$$ for any $$\delta > 0$$.

3.5.2. Show that the sine integral function has the following expansion

3.5.3. Let $$L_{1}^{1}(\mathbf{T}^{1})$$ be the space of all differentiable functions on $$\mathbf{T}^{1}$$ whose derivatives are integrable. Obtain the inclusions $$L_{1}^{1}(\mathbf{T}^{1}) \subseteq BV (\mathbf{T}^{1}) \subseteq L^{\infty }(\mathbf{T}^{1})$$ as follows:

(a) If $$f \in L_{1}^{1}(\mathbf{T}^{1})$$, then $$\mbox{ Var}(f) \leq \| f'\|_{L^{1}}$$.

(b) If $$f \in BV (\mathbf{T}^{1})$$, then $$\|f\|_{L^{\infty }} \leq \mbox{ Var}(f) + \vert f(0)\vert $$.

3.5.4. (a) Let a k  ≥ 0, $$s_{N} =\sum _{ k=-N}^{N}a_{k}$$, and $$\sigma _{N} = \frac{1} {N+1}(\sigma _{0} + \cdots +\sigma _{N})$$. Suppose that $$\sigma _{N} \rightarrow L < \infty $$ as $$N \rightarrow \infty $$. Prove that $$s_{N} \rightarrow L$$ as $$N \rightarrow \infty $$.

(b) Apply the preceding result to show that if a complex-valued function h on $$\mathbf{T}^{1}$$ is continuous in a neighborhood of 0 and $$\widehat{h}(m) \geq 0$$ for all $$m \in \mathbf{Z}$$, then h(0) ≥ 0 and $$\sum _{m\in \mathbf{Z}}\widehat{h}(m) = h(0) < \infty $$; i.e., the partial sums of the Fourier series of h converge at zero.

3.5.5. Let $$h \in L^{1}(\mathbf{T}^{1})$$, $$t_{0} \in \mathbf{T}^{1}$$, and $$0 <\delta < 1/2$$.

(a) Show that $$(h {\ast} D_{N})(t_{0}) \rightarrow L$$ as $$N \rightarrow \infty $$ if and only if

(b) Conclude that if an integrable function h on $$\mathbf{T}^{1}$$ satisfies

then $$(h {\ast} D_{N})(t_{0}) \rightarrow L$$ as $$N \rightarrow \infty $$.

(c) In particular, if there are constants $$C,\beta > 0$$ with $$\beta < 1$$ such that for all t with $$0 < t <\delta$$ we have

then $$(h {\ast} D_{N})(t_{0}) \rightarrow h(t_{0})$$ as $$N \rightarrow \infty $$.

(d) If h is an odd function, then $$(h {\ast} D_{N})(0) \rightarrow 0$$ as $$N \rightarrow \infty $$.

3.5.6. Let $$f \in L^{1}(\mathbf{T}^{1})$$ and suppose that (a, b) is an interval in $$\mathbf{T}^{1}$$. Then we have

[Hint: Use Theorems 3.5.4 and 3.5.5 and the fact that the operator $$f\mapsto f {\ast} D_{N}$$ is self-adjoint.]

3.6 Lacunary Series and Sidon Sets

Lacunary series provide examples of 1-periodic functions on the line that possess certain remarkable properties.

3.6.1 Definition and Basic Properties of Lacunary Series

We begin by defining lacunary sequences.

Definition 3.6.1.

A sequence of positive integers $$\varLambda =\{\lambda _{k}\}_{k=1}^{\infty }$$ is called lacunary if there exists a constant A > 1 such that $$\lambda _{k+1} \geq A\lambda _{k}$$ for all $$k \in \mathbf{Z}^{+}$$.

Examples of lacunary sequences are provided by exponential sequences, such as $$\lambda _{k} = 2^{k},3^{k},4^{k},\ldots$$. Observe that polynomial sequences such as $$\lambda _{k} = 1 + k^{2}$$ are not lacunary. Note that lacunary sequences tend to infinity as $$k \rightarrow \infty $$.

An important observation about lacunary sequences is the following: for any $$m,k_{0} \in \mathbf{Z}^{+}$$ we have

(3.6.1)

Indeed, to prove this assertion, notice that the closest numbers to $$\lambda _{k_{0}}$$ among of the terms of the sequence $$\{\lambda _{k}\}_{k=1}^{\infty }$$ are $$\lambda _{k_{0}+1}$$ and $$\lambda _{k_{0}-1}$$ (the latter only if k ₀ > 1) and thus if j > k ₀ we have

while if j < k ₀

Thus (3.6.1) follows.

We begin with the following result.

Proposition 3.6.2.

Let $$\{\lambda _{k}\}_{k=1}^{\infty }$$ be a lacunary sequence and let f be an integrable function on the circle that is differentiable at a point and has Fourier coefficients

(3.6.2)

Then we have

Proof.

Applying translation, we may assume that the point at which f is differentiable is the origin. Replacing f by the 1-periodic function

we may assume that f(0) = f′(0) = 0. (We have $$\widehat{g}(m) =\widehat{ f}(m)$$ for | m | ≥ 2 and thus the final conclusion for f is equivalent to that for g.)

Using (3.6.1) and (3.6.2), we obtain that for any $$m \in \mathbf{Z}$$ we have

(3.6.3)

Let [t] denote the integer part of t. Given $$\varepsilon > 0$$, pick a positive integer k ₀ such that if $$[(1 - A^{-1})\lambda _{k_{0}}] = 2N_{0}$$, then $$N_{0}^{-2} <\varepsilon$$, and

(3.6.4)

The expression in (3.6.4) can be made arbitrarily small, since f is differentiable at the origin. Now take an integer k with k ≥ k ₀ and set $$2N = [\min (A - 1,1 - A^{-1})\lambda _{k}]$$, which is of course at least 2N ₀. Using (3.6.3), we obtain that for any trigonometric polynomial K N of degree 2N with $$\widehat{K_{N}}(0) = 1$$ we have

(3.6.5)

We take $$K_{N} = (F_{N}/\|F_{N}\|_{L^{2}})^{2}$$, where F N is the Fejér kernel. Using (3.1.18), we obtain first the identity

(3.6.6)

and also the estimate

(3.6.7)

which is valid for | x | ≤ 1∕2. In view of (3.6.6) and (3.6.7), we have the estimate

(3.6.8)

We now use (3.6.5) to obtain

where

Since $$\|K_{N}\|_{L^{1}} = 1$$, it follows that

which can be made arbitrarily small if $$\varepsilon$$ is small. Also, using (3.6.8), we obtain

which, as observed, is bounded by a constant multiple of $$\varepsilon$$. Finally, using again (3.6.8), we obtain

It follows that for all $$k \geq k_{0}$$ we have

for some fixed constant C(f). This proves the required conclusion. □ 

Corollary 3.6.3.

(Weierstrass) There exists a continuous function on the circle that is nowhere differentiable.

Proof.

Consider the 1-periodic function

Since this series converges absolutely and uniformly, f is a continuous function. If f were differentiable at a point, then by Proposition 3.6.2 we would have that $$3^{k}\widehat{f}(3^{k})$$ tends to zero as $$k \rightarrow \infty $$. Since $$\widehat{f}(3^{k}) = 2^{-k}$$ for k ≥ 0, this is not the case. Therefore, f is nowhere differentiable. The real and imaginary parts of this function are displayed in Figure 3.4. □ 

Fig. 3.4

The graph of the real and imaginary parts of the function $$f(t) =\sum _{ k=0}^{\infty }2^{-k}e^{2\pi i3^{k}t }$$.

3.6.2 Equivalence of $$L^{p}$$ Norms of Lacunary Series

We now turn to one of the most important properties of lacunary series, equivalence of their norms. It is a remarkable result that lacunary Fourier series have comparable $$L^{p}$$ norms for $$1 \leq p < \infty $$. More precisely, we have the following theorem:

Theorem 3.6.4.

Let $$1 \leq \lambda _{1} <\lambda _{2} <\lambda _{3} < \cdots $$ be a lacunary sequence with constant A > 1. Set $$\varLambda =\{\lambda _{k}:\,\, k \in \mathbf{Z}^{+}\}$$ . Then for all $$1 \leq p < \infty $$ , there exists a constant C p (A) such that for all $$f \in L^{1}(\mathbf{T}^{1})$$ , with $$\widehat{f}(k) = 0$$ , when $$k \in \mathbf{Z}\setminus \varLambda$$ we have

(3.6.9)

Moreover, the converse inequality to (3.6.9) is valid, and thus all $$L^{p}$$ norms of lacunary Fourier series are equivalent for $$1 \leq p < \infty $$ .

Proof.

We suppose initially that $$f \in L^{2}(\mathbf{T}^{1})$$ and f is nonzero. We define

(3.6.10)

Given $$2 \leq p < \infty $$, we pick an integer m with 2m > p and we also pick a positive integer r such that A r  > m. Then we can write f N as a sum of r functions $$\varphi _{s}$$, $$s = 1,2,\ldots,r$$, where each $$\varphi _{s}$$ has Fourier coefficients that vanish except possibly on the lacunary set

It is a simple fact that the sequence $$\{\mu _{k}\}_{k}$$ is lacunary with constant A r . Then we have

We claim that if $$\mu _{j_{1}} +\ldots +\mu _{j_{m}} =\mu _{k_{1}} +\ldots +\mu _{k_{m}}$$, then

Indeed, if $$\max (\mu _{j_{1}},\ldots,\mu _{j_{m}}) >\max (\mu _{k_{1}},\ldots,\mu _{k_{m}})$$, then

But since

it would follow that $$A^{r} \leq m$$, which contradicts our choice of r. Likewise, we eliminate the case $$\max (\mu _{j_{1}},\ldots,\mu _{j_{m}}) <\max (\mu _{k_{1}},\ldots,\mu _{k_{m}})$$. We conclude that these numbers are equal. We can now continue the same reasoning using induction to conclude that if $$\mu _{j_{1}} +\ldots +\mu _{j_{m}} =\mu _{k_{1}} +\ldots +\mu _{k_{m}}$$, then

Using this fact in the evaluation of the previous multiple sum, we obtain

which implies that $$\|\varphi _{s}\|_{L^{2m}} =\|\varphi _{s}\|_{L^{2}}$$ for all $$s \in \{ 1,2,\ldots,r\}$$. Then we have

since the functions $$\varphi _{s}$$ are orthogonal in L ². Since r can be chosen to be $$[\log _{A}m] + 1$$ and m can be taken to be $$[\frac{p} {2}] + 1$$, we have now established the inequality

(3.6.11)

with $$c_{p}(A) = \sqrt{1 +\big [\log _{A } \big([\frac{p} {2}] + 1\big)\big]}$$ for every f N of the form (3.6.10).

To replace f N by f in (3.6.11), we recall our assumption that $$f \in L^{2}(\mathbf{T}^{1})$$. We observe that $$f_{N} \rightarrow f$$ in L ² and thus $$f_{N_{j}}$$ tends to f a.e. for some subsequence. Then Fatou’s lemma and (3.6.11) imply for $$1 < p < \infty $$

We conclude that

(3.6.12)

By interpolation we obtain

We are assuming that $$0 <\| f\|_{L^{2}} < \infty $$ and the preceding inequality implies that

(3.6.13)

Finally an easy consequence of Hölder’s inequality is that

(3.6.14)

Combining (3.6.12) and (3.6.14) with (3.6.13) yields (3.6.9) with

for all $$1 \leq p < \infty $$ under the hypothesis that $$\widehat{f}(k) = 0$$ for all $$k \in \mathbf{Z}\setminus \varLambda$$ and the additional assumption that $$f \in L^{2}$$.

We now extend the result to $$f \in L^{1}(\mathbf{T}^{1})$$. Given $$f \in L^{1}(\mathbf{T}^{1})$$ with $$\widehat{f}(k) = 0$$ when $$k \in \mathbf{Z}\setminus \varLambda$$, consider the functions f ∗ F M , where F M is the Fejér kernel and $$M \in \mathbf{Z}^{+}$$. Then f ∗ F M lie in L ², f ∗ F M converge to f in L ¹ and in L p , and $$\widehat{f {\ast} F_{M}}(k) = 0$$ when $$k \in \mathbf{Z}\setminus \varLambda$$. The inequality

(3.6.15)

holds since f ∗ F M lie in L ², so letting $$M \rightarrow \infty $$ yields (3.6.9). □ 

Theorem 3.6.4 describes the equivalence of the $$L^{p}$$ norms of lacunary Fourier series for $$p < \infty $$. The question that remains is whether there is a similar characterization for the $$L^{\infty }$$ norms of lacunary Fourier series. Such a characterization is investigated below. Before we state and prove this theorem, we need a classical tool, referred to as a Riesz product.

Definition 3.6.5.

A Riesz product is a function of the form

(3.6.16)

where N is a positive integer, $$\lambda _{1} <\lambda _{2} <\ldots <\lambda _{N}$$ is a lacunary sequence of positive integers, a j are real numbers in [−1, 1], and $$\gamma _{j} \in [0,1]$$.

We make a few observations about Riesz products. A simple calculation gives that if $$P_{N,j}(x) = 1 + a_{j}\cos (2\pi \lambda _{j}x + 2\pi \gamma _{j})$$, then

(3.6.17)

Assume that the constant A associated with the lacunary sequence $$\lambda _{1}\! <\!\lambda _{2}\! <\! \cdots \! <\!\lambda _{N}$$ satisfies A ≥ 3. Then each integer m has at most one representation as a sum

where $$\varepsilon _{j} \in \{-1,1,0\}$$; see Exercise 3.6.1. We now calculate the Fourier coefficients of the Riesz product defined in (3.6.16). For a fixed integer b, let us denote by $$\delta _{b}$$ the sequence of integers that is equal to 1 at b and zero otherwise. Then, using (3.6.17), we obtain that

and thus $$\widehat{P_{N}}$$ is the N-fold convolution of these functions. Using that $$\delta _{a} {\ast}\delta _{b} =\delta _{a+b}$$, we obtain

It follows that $$\widehat{P_{N}}(\lambda _{j}) = 0$$ since for $$j \geq N + 1$$, $$\lambda _{j}$$ cannot be expressed as a linear combination of $$\lambda _{1},\ldots,\lambda _{N}$$ with coefficients in $$\{\pm 1,0\}$$. Also $$\widehat{P_{N}}(\lambda _{k}) = \frac{1} {2}a_{j}e^{2\pi i\gamma _{k}}$$ for $$1 \leq k \leq N$$, since each $$\lambda _{k}$$ is written uniquely as $$0 \cdot \lambda _{1} +\ldots +0 \cdot \lambda _{k-1} + 1 \cdot \lambda _{k}$$. Hence when $$A \geq 3$$ we have that $$\widehat{P_{N}}(\lambda _{k}) = \frac{1} {2}a_{j}e^{2\pi i\gamma _{k}}$$ when 1 ≤ k ≤ N and $$\widehat{P_{N}}(\lambda _{k}) = 0$$ for k ≥ N + 1.

Next, we discuss an important property of Riesz products. Suppose that for some $$m \in \mathbf{Z}$$ we have $$\widehat{P_{N}}(m)\neq 0$$. We write $$m =\sum _{ j=1}^{N}\varepsilon _{j}\lambda _{j}$$ uniquely with $$\varepsilon _{j} \in \{-1,0,1\}$$. Let k be the largest integer less than or equal to N such that $$\varepsilon _{k}\neq 0$$. Then we have

(3.6.18)

Another important property of the Riesz product is that since $$P_{N} \geq 0$$ we have

We recall the space $$A(\mathbf{T}^{1})$$ of all functions with absolutely summable Fourier coefficients normed with the ℓ ¹ norm of the coefficients.

Theorem 3.6.6.

Let $$1 <\lambda _{1} <\lambda _{2} <\lambda _{3} < \cdots $$ be a lacunary sequence of integers with constant A > 1. Set $$\varLambda =\{\lambda _{k}:\,\, k \in \mathbf{Z}^{+}\}$$ . Then there exists a constant C(A) such that for all $$f \in L^{\infty }(\mathbf{T}^{1})$$ with $$\widehat{f}(k) = 0$$ when $$k \in \mathbf{Z}\setminus \varLambda$$ we have

(3.6.19)

Proof.

Let us assume first that $$A \geq 3$$. Also fix $$f \in L^{\infty }(\mathbf{T}^{1})$$. We consider the Riesz product

where $$\gamma _{j}$$ is chosen to satisfy the identity $$\vert \widehat{f}(\lambda _{j})\vert = e^{2\pi i\gamma _{j}}\overline{\widehat{f}(\lambda _{j})}$$. In view of Parseval’s relation and of the fact that $$\|P_{N}\|_{L^{1}} = 1$$ we obtain

(3.6.20)

and the sum in (3.6.20) is finite, since the Fourier coefficients of $$\widehat{P_{N}}$$ form a finitely supported sequence. But $$\widehat{f}(m) = 0$$ for $$m\notin \varLambda$$, while $$\widehat{P_{N}}(\lambda _{j}) = \frac{1} {2}e^{2\pi i\gamma _{j}}$$ for $$1 \leq j \leq N$$ since $$A \geq 3$$, and moreover, $$\widehat{P_{N}}(\lambda _{j}) = 0$$ for j ≥ N + 1, as observed earlier. Thus (3.6.20) reduces to

Letting $$N \rightarrow \infty $$, we deduce that $$\sum _{j=1}^{\infty }\vert \widehat{f}(\lambda _{j})\vert \leq 2\|f\|_{L^{\infty }}$$, which proves (3.6.19) when $$A \geq 3$$.

We now consider the case A < 3. We fix 1 < A < 3 and we pick a positive integer r such that

(3.6.21)

This is possible, since $$(A^{r} - 1)^{-1} \rightarrow 0$$ as $$r \rightarrow \infty $$.

For each $$s \in \{ 1,\ldots,r\}$$, define the sequences $$\lambda _{k}^{s} =\lambda _{s+(k-1)r}$$ indexed by $$k = 1,2,3,\ldots$$ and observe that $$\lambda _{(k+1)}^{s} > A^{r}\lambda _{k}^{s}$$ for all $$k = 1,2,\ldots$$; i.e., each such sequence is lacunary with constant A r . We consider the Riesz product

where $$\gamma _{k}^{s}$$ is defined via the identity $$\vert \widehat{f}(\lambda _{k}^{s})\vert = e^{2\pi i\gamma _{k}^{s} }\,\overline{\widehat{f}(\lambda _{k}^{s})}$$.

Using (3.6.18) we obtain that, if $$m \in \mathbf{Z}$$ is such that $$\widehat{P_{N}^{s}}(m)\neq 0$$, then there exists a $$k \in \{ 1,2\,\ldots,N\}$$ such that

This combined with (3.6.21) yields

Using (3.6.1) we obtain that either $$m = \pm \lambda _{k}^{s}$$ or $$\vert m\vert \notin \varLambda$$. Thus we have

This observation, the fact that $$\widehat{f}$$ is supported in $$\varLambda$$, and Parseval’s relation yield

(3.6.22)

Since $$\widehat{P_{N}}(\lambda _{k}^{s}) = \frac{1} {2}e^{2\pi i\gamma _{k}^{s} }$$ for 1 ≤ k ≤ N, (3.6.22) reduces to

Letting $$N \rightarrow \infty $$ gives

Summing over s in the set $$\{1,2,\ldots,r\}$$, we obtain the required conclusion with C(A) = 2r and note that r can be taken to be $$[\max (\log _{A}\frac{2A-1} {A-1},\log _{3}A)] + 2$$. □ 

Corollary 3.6.7.

Let $$\varLambda =\{\lambda _{k}:\,\, k \in \mathbf{Z}^{+}\}$$ be a lacunary set and let f be a bounded function on the circle that satisfies $$\widehat{f}(k) = 0$$ when $$k \in \mathbf{Z}\setminus \varLambda$$ . Then f is almost everywhere equal to the absolutely (and uniformly) convergent series

(3.6.23)

and thus it is almost everywhere equal to a continuous function.

Proof.

It follows from Theorem 3.6.6 that if $$\widehat{f}(k) = 0$$ when $$k \in \mathbf{Z}\setminus \varLambda$$, then we have that $$f \in A(\mathbf{T}^{1})$$. Applying the inversion result in Proposition 3.2.5 we obtain that f is almost everywhere equal to a continuous function and that (3.6.23) holds for almost all $$x \in \mathbf{T}^{1}$$. □ 

3.6.3 Sidon sets

Given a subset E of the integers, we denote by $$\mathcal{C}_{E}$$ the space of all continuous functions on $$\mathbf{T}^{1}$$ such that

(3.6.24)

It is straightforward that $$\mathcal{C}_{E}$$ is a closed subspace of all bounded functions on the circle $$\mathbf{T}^{1}$$ with the standard $$L^{\infty }$$ norm.

Definition 3.6.8.

A set of integers E is called a Sidon set if every function in $$\mathcal{C}_{E}$$ has an absolutely convergent Fourier series.

There are several characterizations of Sidon sets. We state them below.

Proposition 3.6.9.

The following assertions are equivalent for a subset E of $$\mathbf{Z}$$.

(1)

There is a constant K such that for all trigonometric polynomials P with $$\widehat{P}$$ supported in E we have

(2)

There exists a constant K such that

for every bounded function f on $$\mathbf{T}^{1}$$ with $$\widehat{f}$$ supported in E.

(3)

Every function f in $$\mathcal{C}_{E}$$ has an absolutely convergent Fourier series; i.e., E is a Sidon set.

(4)

For every bounded function b on E there is a finite Borel measure μ on $$\mathbf{T}^{1}$$ such that $$\widehat{\mu }(m) = b(m)$$ for all m ∈ E.

(5)

For every function b on $$\mathbf{Z}$$ with the property $$b(m) \rightarrow 0$$ as $$m \rightarrow \infty $$ , there is a function $$g \in L^{1}(\mathbf{T}^{1})$$ such that $$\widehat{g}(m) = b(m)$$ for all $$m \in E$$ .

Proof.

Suppose that (1) holds. Given f in $$L^{\infty }(\mathbf{T}^{1})$$ with $$\widehat{f}$$ is supported in E, write

where F N is the Fejér kernel. These are trigonometric polynomials whose Fourier coefficients vanish on $$\mathbf{Z}\setminus E$$. Applying (1) we obtain

Letting $$N \rightarrow \infty $$ we obtain (2).

It is trivial that (2) implies (3).

If (3) holds, then the map $$f\mapsto \widehat{f}$$ is a linear bijection from $$\mathcal{C}_{E}$$ to $$\ell^{1}(E)$$. Moreover its inverse mapping $$\widehat{f}\mapsto f$$ is continuous, since

By the open mapping theorem, it follows that $$f\mapsto \widehat{f}$$ is a continuous mapping, which proves the existence of a constant K such that (1) holds.

We have now proved the equivalence of (1), (2), and (3).

We show that (2) implies (4). If E is a Sidon set and if b is a bounded function on E, say $$\|b\|_{\ell^{\infty }} \leq 1$$, then the mapping

is a bounded linear functional on $$\mathcal{C}_{E}$$ with norm at most K. By the Hanh-Banach theorem this functional admits an extension to $$\mathcal{C}(\mathbf{T}^{1})$$ with the same norm. Hence there is a measure μ, whose total variation $$\|\mu \|$$ does not exceed K, such that

Taking $$f(t) = e^{2\pi imt}$$ in (2) we obtain $$\widehat{\mu }(m) = b(m)$$ for all $$m \in E$$.

If (4) holds and $$b(m) \rightarrow 0$$ as $$\vert m\vert \rightarrow \infty $$, using Lemma 3.3.2 there is a convex sequence c(m) such that c(m) > 0, c(m) → 0 as $$\vert m\vert \rightarrow \infty $$, c(−m) = c(m), and $$\vert b(m)\vert \leq c(m)$$ for all $$m \in \mathbf{Z}$$. By (4), there is a finite Borel measure μ with $$\widehat{\mu }(m) = b(m)/c(m)$$ for all $$m \in E$$.

By Theorem 3.3.4, there is a function g in $$L^{1}(\mathbf{T}^{1})$$ such that $$\widehat{g}(m) = c(m)$$ for all $$m \in \mathbf{Z}$$. Then $$b(m) =\widehat{ g}(m)\widehat{\mu }(m)$$ for all m ∈ E. Since $$f = g{\ast}\mu$$ is in L ¹, we have $$b(m) =\widehat{ f}(m)$$ for all $$m \in E$$, and thus (4) implies (5).

Finally, if (5) holds, we show (3). Given $$f \in \mathcal{C}_{E}$$, we show that for an arbitrary sequence d m tending to zero, we have $$\sum _{m\in \mathbf{Z}}\vert \widehat{f}(m)d_{m}\vert < \infty $$; this implies that $$\sum _{m\in \mathbf{Z}}\vert \widehat{f}(m)\vert < \infty $$. Given a sequence $$d_{m} \rightarrow 0$$, pick a function g in L ¹ such that $$\widehat{g}(m)\widehat{f}(m) = \vert \widehat{f}(m)\vert \,\vert d_{m}\vert $$ for all $$m \in E$$ by assumption (5). Then the series

(3.6.25)

has nonnegative terms and the function f ∗ g is continuous, thus $$F_{N} {\ast} (f {\ast} g)(0) \rightarrow (f {\ast} g)(0)$$ as $$N \rightarrow \infty $$. It follows that $$D_{N} {\ast} (f {\ast} g)(0) \rightarrow (f {\ast} g)(0)$$, thus the series in (3.6.25) converges (see Exercise 3.5.4) and hence $$\sum _{m\in \mathbf{Z}}\vert \widehat{f}(m)d_{m}\vert < \infty.$$ □ 

Example 3.6.10.

Every lacunary set is a Sidon set. Indeed, suppose that E is a lacunary set with constant A. If f is a continuous function which satisfies (3.6.24), then Theorem 3.6.6 gives that

hence f has an absolutely convergent Fourier series.

Example 3.6.11.

There exist subsets of $$\mathbf{Z}$$ that are not Sidon. For example, $$\mathbf{Z}\setminus \{0\}$$ is not a Sidon set. See Exercise 3.6.7.

Exercises

3.6.1. Suppose that $$0 <\lambda _{1} <\lambda _{2} <\ldots <\lambda _{N}$$ is a lacunary sequence of integers with constant $$A \geq 3$$. Prove that for every integer m there exists at most one N-tuple $$(\varepsilon _{1},\ldots,\varepsilon _{N})$$ with each $$\varepsilon _{j} \in \{-1,1,0\}$$ such that

[Hint: Suppose there exist two such N-tuples. Pick the largest k such that the coefficients of $$\lambda _{k}$$ are different.]

3.6.2. Is the sequence $$\lambda _{k} =\big [e^{(\log k)^{2} }\big]$$, $$k = 2,3,4,\ldots$$ lacunary?

3.6.3. Let $$a_{k} \geq 0$$ for all $$k \in \mathbf{Z}^{+}$$ and $$1 \leq p < \infty $$. Show that there exist constants C p , c p such that for all $$N \in \mathbf{Z}^{+}$$ we have

while

3.6.4. Suppose that $$0 <\lambda _{1} <\lambda _{2} < \cdots $$ is a lacunary sequence and let f be a bounded function on the circle that satisfies $$\widehat{f}(m) = 0$$ whenever $$m \in \mathbf{Z}\setminus \{\lambda _{1},\lambda _{2},\ldots \}$$. Suppose also that

for some $$0 <\alpha < 1$$.

(a) Prove that there is a constant C such that $$\vert \widehat{f}(\lambda _{k})\vert \leq CB\lambda _{k}^{-\alpha }$$ for all k ≥ 1.

(b) Prove that $$f \in \dot{\varLambda }_{\alpha }(\mathbf{T}^{1}).$$

[Hint: Let $$2N = [(1 - A^{-1})\lambda _{k}]$$ and let K N be as in the proof of Proposition 3.6.2. Write

Use that $$\|K_{N}\|_{L^{1}} = 1$$ and also the estimate (3.6.7). Part (b): Use the estimate in part (a).]

3.6.5. Let f be an integrable function on the circle whose Fourier coefficients vanish outside a lacunary set $$\varLambda =\{\lambda _{1},\lambda _{2},\lambda _{3},\ldots \}$$. Suppose that f vanishes identically in a small neighborhood of the origin. Show that f is in $$\mathcal{C}^{\infty }(\mathbf{T}^{1})$$.

[Hint: Let $$2N = [(1 - A^{-1})\lambda _{k}]$$ and let K N be as in the proof of Proposition 3.6.2. Write

and use estimate (3.6.7) to obtain that f is in $$\mathcal{C}^{2}$$. Continue by induction.]

3.6.6. Let $$1 < a,b < \infty $$. Consider the 1-periodic function

Prove that the following statements are equivalent:

(a) f is differentiable at a point.

(b) b < a.

(c) f is differentiable everywhere.

3.6.7. Use the example in Proposition 3.4.6 (a) to show that $$\mathbf{Z}\setminus \{q_{1},\ldots,q_{L}\}$$ is not a Sidon set for any finite subset $$\{q_{1},\ldots,q_{L}\}$$ of the integers.

3.6.8. Let $$0 <\delta < 1$$. Let E be a subset of the integers such that for any sequence of complex numbers $$\{d_{m}\}_{m\in E}$$ with | d m  |  = 1 there is a finite Borel measure $$\mu$$ on $$\mathbf{T}^{1}$$ such that

for all m ∈ E. Show that E is a Sidon set.

[Hint: Given f be in $$\mathcal{C}_{E}$$ define d m via the identity $$d_{m}\widehat{f}(m) = \vert \widehat{f}(m)\vert $$ if $$\widehat{f}(m)\neq 0$$, otherwise set d m  = 1. For the measure μ given by the hypothesis, notice that $$\mbox{ Re}\,(\widehat{\mu }(m)\widehat{f}(m)) \geq \delta \vert \widehat{f}(m)\vert $$ for all $$m \in \mathbf{Z}$$.]

HISTORICAL NOTES

Trigonometric series in one dimension were first considered in the study of the vibrating string problem and are implicitly contained in the work of d’Alembert, D. Bernoulli, Clairaut, and Euler. The analogous problem for vibrating higher-dimensional bodies naturally suggested the use of multiple trigonometric series. However, it was the work of Fourier on steady-state heat conduction that inspired the subsequent systematic development of such series. Fourier announced his results in 1811, although his classical book Théorie de la chaleur was published in 1822. This book contains several examples of heuristic use of trigonometric expansions and motivated other mathematicians to carefully study such expansions. The systematic development of the theory of Fourier series began by Dirichlet [96], who studied the pointwise convergence of the Fourier series of piecewise monotonic functions via the use of the kernel D N , today called the Dirichlet kernel.

The fact that the Fourier series of a continuous function can diverge was first observed by DuBois Reymond in 1873. The Riemann–Lebesgue lemma was first proved by Riemann in his memoir on trigonometric series (appeared between 1850 and 1860). It carries Lebesgue’s name today because Lebesgue later extended it to his notion of integral. The rebuilding of the theory of Fourier series based on Lebesgue’s integral was mainly achieved by de la Vallée-Poussin and Fatou.

Theorem 3.3.16 was obtained by Bernstein [26] in dimension n = 1. Higher-dimensional analogues of the Hardy–Littlewood series of Exercise 3.3.8 were studied by Wainger [370]. These series can be used to produce examples indicating that the restriction $$s >\alpha +n/2$$ in Bernstein’s theorem is sharp even in higher dimensions. Part (b) of Theorem 3.4.4 is due to Lebesgue when n = 1 and Marcinkiewicz and Zygmund [243] when n = 2. Marcinkiewicz and Zygmund’s proof also extends to higher dimensions. The proof given here is based on Lemma 3.4.5 proved by Stein [342] in a different context. The proof of Lemma 3.4.5 presented here was suggested by T. Tao.

Abel proved that if an infinite series $$\sum _{k=0}^{\infty }a_{k}$$ converges and has sum L, then the power series $$f(x) =\sum _{ k=0}^{\infty }a_{k}x^{k}$$ converges for | x |  < 1 and tends to L as $$x \rightarrow 1-$$. The converse of this theorem under the additional assumption that $$ka_{k} \rightarrow 0$$ as $$k \rightarrow \infty $$ was proved by Tauber [358]. Hardy [143] extended Tauber’s result (Theorem 3.5.1) for Cesàro summability under the weaker assumption that the sequence ka k is bounded. Jordan [180] studied of functions of bounded variation and proved Theorem 3.5.4. The existence of a continuous function which is nowhere differentiable (Corollary 3.6.3) was first published in 1872 by K. Weierstrass, although earlier findings of such functions were published later. The exposition on Sidon sets is taken from the classical article of Rudin [305], which also contains Exercise 3.6.8.

The Gibbs phenomenon (a version of Theorem 3.5.7) was discovered by Wilbraham [376] and rediscovered by Gibbs [128]; this phenomenon describes the particular way in which the Fourier sums of a piecewise continuously differentiable periodic function have large oscillations and overshoot at the jump discontinuity of the function. Bôcher [29] gave a detailed mathematical analysis of that overshoot, which he called the “Gibbs phenomenon”.

The main references for trigonometric series are the books of Bary [20] and Zygmund [388], [389]. Other references for one-dimensional Fourier series include the books of Edwards [106], Dym and McKean [105], Katznelson [190], Körner [202], Pinsky [283], and the first eight chapters in Torchinsky [363]. The reader may also consult the book of Krantz [203] for a historical introduction to the subject of Fourier series. A review of the heritage and continuing significance of Fourier Analysis is written by Kahane [182].

A classical treatment of multiple Fourier series can be found in the last chapter of Bochner’s book [32] and in parts of his other book [31]. Other references include the last chapter in Zygmund [389], the books of Yanushauskas [381] (in Russian) and Zhizhiashvili [384], the last chapter in Stein and Weiss [348], and the article of Alimov, Ashurov, and Pulatov in [3]. A brief survey article on the subject was written by Ash [11]. More extensive expositions were written by Shapiro [320], Igari [171], and Zhizhiashvili [383]. A short note on the history of Fourier series was written by Zygmund [390]. The book of Shapiro [321] contains a very detailed study of Fourier series in several variables as well as applications of this theory.

References

3.

Alimov, Sh. A., Ashurov, R. R., Pulatov, A. K., Multiple Fourier series and Fourier integrals, Proc. Commutative Harmonic Analysis, IV, pp. 1–95, Encyclopaedia Math. Sci. 42, Springer, Berlin, 1992.

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Ash, J. M., Multiple trigonometric series, Studies in Harmonic Analysis (Proc. Conf., DePaul Univ., Chicago, IL, 1974), pp. 76–96, MAA Stud. Math., Vol. 13, Math. Assoc. Amer., Washington, D. C., 1976.

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Bary, N., A Treatise on Trigonometric Series, Vols. I, II, Authorized translation by Margaret F. Mullins, A Pergamon Press Book, The Macmillan Co., New York, 1964.MATH

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Bernstein, S., Sur la convergence absolue des séries trigonométriques, Comptes Rendus des Séances de l’ Académie des Sciences, Paris, 158 (1914), 1661–1663.MATH

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Bôcher, M., Introduction to the theory of Fourier’s series, Ann. of Math. (2nd Ser.) 7 (1906), no. 3, 81–152.

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Bochner, S., Harmonic Analysis and the Theory of Probability, University of California Press, Berkeley and Los Angeles, 1955.MATH

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Bochner, S., Lectures on Fourier Integrals, (with an author’s supplement on monotonic functions, Stieltjes integrals, and harmonic analysis), Translated by Morris Tenenbaum and Harry Pollard. Annals of Mathematics Studies, No. 42, Princeton University Press, Princeton, NJ, 1959.

96.

Dirichlet, P. G., Sur la convergence des séries trigonométriques qui servent à représenter une fonction arbitraire entre des limites données, J. Reine und Angew. Math. 4 (1829), 157–169.CrossRefMATH

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Dym, H., McKean, H. P., Fourier Series and Integrals, Probability and Mathematical Statistics, No. 14, Academic Press, New York-London, 1972.

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Edwards, R. E., Fourier Series: A Modern Introduction. Vol. 1, 2nd edition, Graduate Texts in Mathematics, 64, Springer-Verlag, New York-Berlin, 1979.

128.

Gibbs, J. W., Fourier’s Series, Nature 59 (1899), 606.CrossRef

143.

Hardy, G. H., Theorems relating to the summability and convergence of slowly oscillating series, Proc. London Math. Soc. (2) 8 (1910), no. 1, 301–320.

171.

Igari, S., Lectures on Fourier Series of Several Variables, Univ. Wisconsin Lecture Notes, Madison, WI, 1968.

180.

Jordan, C., Sur la série de Fourier, Comptes Rendus Hebdomadaires des Séances de l’ Académie des Sciences 92 (1881), 228–230.MATH

182.

Kahane, J.-P., The heritage of Fourier, Perspectives in Analysis, 83–95, Math. Phys. Stud., 27, Springer, Berlin, 2005.

190.

Katznelson, Y., An Introduction to Harmonic Analysis, 2nd corrected edition, Dover Publications, Inc., New York, 1976.

202.

Körner, T. W., Fourier Analysis, Cambridge University Press, Cambridge, 1988.CrossRefMATH

203.

Krantz, S. G., A panorama of Harmonic Analysis, Carus Mathematical Monographs, 27, Mathematical Association of America, Washington, DC, 1999.

243.

Marcinkiewicz, J., Zygmund, A., On the summability of double Fourier series, Fund. Math. 32 (1939), 122–132.

283.

Pinsky, M., Introduction to Fourier Analysis and Wavelets, Graduate Studies in Mathematics 102, American Mathematical Society, Providence, RI, 2009.

305.

Rudin, W., Trigonometric series with gaps, J. Math. Mech. 9 (1960), 203–227.MATHMathSciNet

307.

Rudin, W., Real and Complex Analysis, 2nd edition, Tata McGraw-Hill Publishing Company, New Delhi, 1974.MATH

320.

Shapiro, V. L., Fourier series in several variables, Bull. Amer. Math. Soc. 70 (1964), 48–93.CrossRefMATHMathSciNet

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Shapiro, V. L., Fourier series in several variables with applications to partial differential equations, Chapman & Hall/CRC Applied Mathematics and Nonlinear Science Series, CRC Press, Boca Raton, FL, 2011.CrossRefMATH

342.

Stein, E. M., Boundary behavior of harmonic functions on symmetric spaces: Maximal estimates for Poisson integrals, Invent. Math. 74 (1983), no. 1, 63–83.CrossRefMATHMathSciNet

348.

Stein, E. M., Weiss, G., Introduction to Fourier Analysis on Euclidean Spaces, Princeton Mathematical Series, No. 32, Princeton University Press, Princeton, NJ, 1971.

358.

Tauber, A., Ein Satz aus der Theorie der unendlichen Reihen Monatsh. Math. Phys. 8 (1897), no. 1, 273–277.CrossRefMATHMathSciNet

363.

Torchinsky, A., Real-Variable Methods in Harmonic Analysis, Pure and Applied Mathematics, 123, Academic Press, Inc., Orlando, FL, 1986.

370.

Wainger, S., Special trigonometric series in k dimensions, Mem. Amer. Math. Soc. 59, 1965.

376.

Wilbraham, H., On a certain periodic function, The Cambridge and Dublin Mathematical Journal, 3 (1848), 198–201.

381.

Yanushauskas, A. I., Multiple trigonometric series [Russian], ”Nauka” Sibirsk. Otdel., Novosibirsk, 1986.

383.

Zhizhiashvili, L. V., Some problems in the theory of simple and multiple trigonometric and orthogonal series, Russian Math. Surveys, 28 (1973), 65–127.CrossRef

384.

Zhizhiashvili, L. V., Trigonometric Fourier Series and Their Conjugates, Mathematics and Its Applications, 372, Kluwer Academic Publishers Group, Dordrecht,1996.

388.

Zygmund, A., Trigonometric Series, Vol. I, 2nd edition, Cambridge University Press, New York, 1959.

389.

Zygmund, A., Trigonometric Series, Vol. II, 2nd edition, Cambridge University Press, New York, 1959.

390.

Zygmund, A., Notes on the history of Fourier series, Studies in harmonic analysis, (Proc. Conf. DePaul Univ., Chicago, IL, 1974) pp. 1–19, MAA Stud. Math., Vol. 13, Math. Assoc. Amer., Washington, DC, 1976.

© Springer Science+Business Media New York 2014

Loukas GrafakosClassical Fourier AnalysisGraduate Texts in Mathematics24910.1007/978-1-4939-1194-3_4

4. Topics on Fourier Series

Loukas Grafakos1 

(1)

Department of Mathematics, University of Missouri, Columbia, MO, USA

In this chapter we go deeper into the theory of Fourier series and we study topics such as convergence in norm and the conjugate function, divergence of Fourier series and Bochner–Riesz summability. We also study transference of multipliers on the torus and of maximal multipliers. This is a powerful technique that allows one to infer results concerning Fourier series from corresponding results about Fourier integrals and vice versa.

We also take a quick look at applications of Fourier series such as the isopertimetric inequality problem, the distribution of lattice points in a ball, and the heat equation. The power of Fourier series techniques manifests itself in the study of these problems which represent only a small part of the wide and vast range of applications of the subject known today.

4.1 Convergence in Norm, Conjugate Function, and Bochner–Riesz Means

In this section we address the following fundamental question: Do Fourier series converge in norm? We begin with some abstract necessary and sufficient conditions that guarantee such a convergence. In one dimension, we are able to reduce matters to the study of the so-called conjugate function on the circle, a sister operator of the Hilbert transform, which is the center of study of the next chapter. In higher dimensions the situation is more complicated, but we are able to give a positive answer in the case of square summability.

4.1.1 Equivalent Formulations of Convergence in Norm

The question we pose is for which indices p, with $$1 \leq p < \infty $$, we have

(4.1.1)

and similarly for the circular Dirichlet kernel $$ D_{N }^{n }$$. We tackle this question by looking at an equivalent formulation of it.

Theorem 4.1.1.

For R > 0 and $$m \in \mathbf{Z}^{n}$$ , let a(m,R) be complex numbers such that

(i)

For every R > 0 there is a q R such that a(m,R) = 0 if |m| > q R .

(ii)

There is an $$M_{0} < \infty $$ such that |a(m,R)|≤ M ₀ for all $$m \in \mathbf{Z}^{n}$$ and all R > 0.

(iii)

For each $$m \in \mathbf{Z}^{n}$$ , the limit of a(m,R) exists as $$R \rightarrow \infty $$ and $$\lim _{R\rightarrow \infty }a(m,R) = a_{m}$$ .

Let $$1 \leq p < \infty $$ . For $$f \in L^{p}(\mathbf{T}^{n})$$ and $$x \in \mathbf{T}^{n}$$ define

noting that the sum is well defined because of (i). Also, for $$h \in \mathcal{C}^{\infty }(\mathbf{T}^{n})$$ define

Then for all $$f \in L^{p}(\mathbf{T}^{n})$$ the sequence S R (f) converges in $$L^{p}$$ as $$R \rightarrow \infty $$ if and only if there exists a constant $$K < \infty $$ such that

(4.1.2)

Furthermore, if (4.1.2) holds, then for the same constant K we have

(4.1.3)

and then A extends to a bounded operator $$\widetilde{A}$$ from $$L^{p}(\mathbf{T}^{n})$$ to itself; moreover, for every $$f \in L^{p}(\mathbf{T}^{n})$$ we have that $$S_{R}(f) \rightarrow \widetilde{ A}(f)$$ in $$L^{p}$$ as $$R \rightarrow \infty $$ .

Proof.

If S R (f) converges in $$L^{p}$$, then $$\|S_{R}(f)\|_{L^{p}} \leq C_{f}$$ for some constant C f that depends on $$f \in L^{p}(\mathbf{T}^{n})$$. Moreover, each S R is a bounded operator from $$L^{p}(\mathbf{T}^{n})$$ to itself with norm at most $$\,\#\{m \in \mathbf{Z}^{n}:\,\, \vert m\vert \leq q_{R}\}\,M_{0}.$$ Thus $$\{S_{R}\}_{R>0}$$ is a family of L p -bounded linear operators that satisfy $$\sup _{R>0}\|S_{R}(f)\|_{L^{p}} \leq C_{f}$$ for each $$f \in L^{p}(\mathbf{T}^{n})$$. The uniform boundedness theorem applies and yields that the operator norms of S R from $$L^{p}$$ to $$L^{p}$$ are bounded uniformly in R. This proves (4.1.2).

Conversely, assume (4.1.2). For $$h \in \mathcal{C}^{\infty }(\mathbf{T}^{n})$$, we have that

in view of property (ii) and of the Lebesgue dominated convergence theorem, since $$\sum _{m\in \mathbf{Z}^{n}}\vert \widehat{h}(m)\vert < \infty $$. Fatou’s lemma now gives

hence (4.1.3) holds. Thus A extends to a bounded operator $$\widetilde{A}$$ on $$L^{p}(\mathbf{T}^{n})$$ by density.

We show that for all $$f \in L^{p}(\mathbf{T}^{n})$$ we have $$S_{R}(f) \rightarrow \widetilde{ A}(f)$$ in $$L^{p}$$ as $$R \rightarrow \infty $$. Fix f in $$L^{p}(\mathbf{T}^{n})$$ and let $$\varepsilon > 0$$ be given. Pick a trigonometric polynomial P satisfying $$\|f - P\|_{L^{p}} \leq \varepsilon$$. Let d be the degree of P. Then there is an R ₀ > 0 such that for all $$R > R_{0}$$ we have

since $$a(m,R) \rightarrow a_{m}$$ for every m with $$\vert m_{1}\vert +\ldots +\vert m_{n}\vert \leq d$$. We deduce that

whenever R > R ₀. Then

for $$R > R_{0}$$. This proves that $$S_{R}(f)$$ converges to $$\widetilde{A}(f)$$ in $$L^{p}$$ as $$R \rightarrow \infty $$. □ 

The most interesting situation arises, of course, when $$a(m,R) \rightarrow a_{m} = 1$$ for all $$m \in \mathbf{Z}^{n}$$. In this case A (and $$\widetilde{A}$$) is the identity operator, and thus we expect the operators $$S_{R}(f)$$ to converge back to f as $$R \rightarrow \infty $$. We should keep in mind the following three examples:

(a) The sequence a(m, R) = 1 when | m j  | ≤ R for all $$j \in \{ 1,2,\ldots,n\}$$ and zero otherwise, in which case the operator S R of Theorem 4.1.1 is

(4.1.4)

(b) The sequence a(m, R) = 1 when $$\vert m\vert \leq R$$ and zero otherwise, in which case the S R of Theorem 4.1.1 is

(4.1.5)

(c) The sequence $$a(m,R) =\big (1 -\tfrac{\vert m\vert ^{2}} {R^{2}} \big)_{+}^{\alpha }$$, for some $$\alpha > 0$$, in which case we denote S R by $$B_{R}^{\alpha }$$.

Definition 4.1.2.

The Bochner–Riesz operator or Bochner–Riesz means of order $$\alpha \geq 0$$ is the operator

(4.1.6)

defined on integrable functions f on $$\mathbf{T}^{n}$$.

Corollary 4.1.3.

Let $$1 \leq p < \infty $$ and $$\alpha \geq 0$$ . Let S R and $$ S_{R }$$ be as in (4.1.4) and (4.1.5) , respectively, and let $$B_{R}^{\alpha }$$ be the Bochner–Riesz means as defined in (4.1.6) . Then

Example 4.1.4.

We investigate the one-dimensional case in some detail. We take n = 1, and we define a(m, N) = 1 for all − N ≤ m ≤ N, and zero otherwise. Then $$S_{N}(f) = S_{N } (f) = D_{N} {\ast} f$$, where D N is the Dirichlet kernel. Clearly, the expressions $$\|S_{N}\|_{L^{p}\rightarrow L^{p}}$$ are bounded above by the L ¹ norm of D N , but this estimation yields a bound that blows up as $$N \rightarrow \infty $$. We later show, via a more delicate argument, that the expressions $$\|S_{N}\|_{L^{p}\rightarrow L^{p}}$$ are uniformly bounded in N when $$1 < p < \infty $$.

This reasoning, however, allows us to deduce that for some function $$g \in L^{1}(\mathbf{T}^{1})$$, $$S_{N}(g)$$ may not converge in L ¹. This is also a consequence of the proof of Theorem 4.2.1; see (4.2.13). Note that since the Fejér kernel F M has L ¹ norm 1, we have

This implies that the expressions $$\|S_{N}\|_{L^{1}\rightarrow L^{1}}$$ are not uniformly bounded in N, and therefore Corollary 4.1.3 gives that for some $$f_{0} \in L^{1}(\mathbf{T}^{1})$$, $$S_{N}(f_{0})$$ does not converge to f ₀ in L ¹.

Although the partial sums of Fourier series fail to convergence in $$L^{1}(\mathbf{T}^{n})$$, it is a consequence of Plancherel’s theorem that they converge in $$L^{2}(\mathbf{T}^{n})$$. More precisely, if $$f \in L^{2}(\mathbf{T}^{n})$$, then

as $$N \rightarrow \infty $$ and the same result is true for $$D_{N}^{n} {\ast} f$$ and for $$B_{R}^{\alpha } {\ast} f$$; for the latter, we apply Theorem 4.1.1, noting that

and thus $$\sup _{R>0}\|B_{R}^{\alpha }\|_{L^{2}\rightarrow L^{2}} \leq 1$$.

Motivated by the preceding discussion for p = 2, it is natural to pose the following question. Can 2 be replaced by p ≠ 2 in the preceding results? This question has an affirmative answer in dimension one for D N . In higher dimensions an interesting dichotomy appears. As a consequence of the one-dimensional result, the square partial sums $$D_{N}^{n} {\ast} f$$ converge in L p to a given f in $$L^{p}(\mathbf{T}^{n})$$, but for the circular partial sums this may not be the case.

We begin the discussion with the one-dimensional situation.

Definition 4.1.5.

For $$f \in \mathcal{C}^{\infty }(\mathbf{T}^{1})$$ define the conjugate function $$\widetilde{f}$$ by

where sgn(m) = 1 for m > 0, − 1 for m < 0, and 0 for m = 0. Also define the Riesz projections P ₊ and P ₋ by

(4.1.7)

(4.1.8)

Observe that $$f = P_{+}(f) + P_{-}(f) +\widehat{ f}(0)$$, while $$\widetilde{f} = -iP_{+}(f) + iP_{-}(f)$$, when f is in $$\mathcal{C}^{\infty }(\mathbf{T}^{1})$$. Consequently, one has

(4.1.9)

and therefore the $$L^{p}$$ boundedness of the operator $$f\mapsto \widetilde{f}$$ is equivalent to that of the operator $$f\mapsto P_{+}(f)$$, since the identity and the operator $$f\mapsto \widehat{f}(0)$$ are obviously $$L^{p}$$ bounded. Clearly, these statements are also valid for the other Riesz projection $$f\mapsto P_{-}(f)$$. The following is a consequence of Theorem 4.1.1.

Proposition 4.1.6.

Let $$1 \leq p < \infty $$ . Then the expressions $$S_{N}(f) = D_{N} {\ast} f$$ converge to f in $$L^{p}(\mathbf{T}^{1})$$ as $$N \rightarrow \infty $$ if and only if there exists a constant C p > 0 such that for all smooth functions f on $$\mathbf{T}^{n}$$ we have $$\|\widetilde{f}\,\|_{L^{p}(\mathbf{T}^{1})} \leq C_{p}\|f\|_{L^{p}(\mathbf{T}^{1})}$$ .

Proof.

In view of Corollary 4.1.3, the fact that for all $$f \in L^{p}(\mathbf{T}^{1})$$, S N (f) → f in $$L^{p}$$ as $$N \rightarrow \infty $$ is equivalent to the uniform (in N) $$L^{p}$$ boundedness of S N .

We note the validity of the identity

Since multiplication by exponentials does not affect $$L^{p}$$ norms, this identity implies that the norm of the operator $$S_{N}(f) = D_{N} {\ast} f$$ from $$L^{p}$$ to $$L^{p}$$ is equal to that of the operator

from $$L^{p}$$ to $$L^{p}$$. Therefore,

(4.1.10)

and both of these statements are equivalent to the fact that for all $$f \in L^{p}(\mathbf{T}^{1})$$, $$S_{N}(f) \rightarrow f$$ in $$L^{p}$$ as $$N \rightarrow \infty $$.

We have already observed that the L p boundedness of the conjugate function is equivalent to that of P ₊. Therefore, it suffices to show that the L p boundedness of P ₊ is equivalent to the uniform L p boundedness of S N ′.

Suppose first that $$\sup _{N\geq 0}\|S_{N}'\|_{L^{p}\rightarrow L^{p}} < \infty $$. Theorem 4.1.1 applied to the sequence a(m, R) = 1 for 0 ≤ m ≤ R and a(m, R) = 0 otherwise gives that the operator $$A(f) = P_{+}(f) +\widehat{ f}(0)$$ is bounded on $$L^{p}(\mathbf{T}^{1})$$. Hence so is P ₊.

Conversely, suppose that P ₊ extends to a bounded operator from $$L^{p}(\mathbf{T}^{1})$$ to itself. For all h in $$\mathcal{C}^{\infty }(\mathbf{T}^{n})$$ we can write

This identity implies that

(4.1.11)

for all f smooth, and by density for all $$f \in L^{p}(\mathbf{T}^{1}).$$ Note that S N ′ is well defined on $$L^{p}(\mathbf{T}^{1})$$. Thus the operators S N ′ are uniformly bounded on $$L^{p}(\mathbf{T}^{n})$$.

Thus the uniform L p boundedness of S N is equivalent to the uniform L p boundedness of S N ′, which is equivalent to the L p boundedness of P ₊, which in turn is equivalent to the L p boundedness of the conjugate function. □ 

4.1.2 The $$L^{p}$$ Boundedness of the Conjugate Function

We know now that convergence of Fourier series in $$L^{p}$$ is equivalent to the $$L^{p}$$ boundedness of the conjugate function or either of the two Riesz projections. It is natural to ask whether these operators are $$L^{p}$$ bounded.

Theorem 4.1.7.

Given $$1 < p < \infty $$ , there is a constant A p > 0 such that for all f in $$\mathcal{C}^{\infty }(\mathbf{T}^{1})$$ we have

(4.1.12)

Thus the operator $$f\mapsto \widetilde{f}$$ has a bounded extension on $$L^{p}(\mathbf{T}^{1})$$ that also satisfies (4.1.12).

Consequently, the Fourier series of $$L^{p}$$ functions on the circle converge back to the functions in the $$L^{p}$$ norm for $$1 < p < \infty $$.

Proof.

In proving the inequality (4.1.12), we make the following reductions:(a) We assume that f is trigonometric polynomial. (b) We assume that $$\widehat{f}(0) = 0$$.(c) We assume that f is real valued.

Since f is a real-valued function, we have that $$\widehat{f}(-m) = \overline{\widehat{f}(m)}$$ for all m, and since $$\widehat{f}(0) = 0$$, we can write

which implies that $$\widetilde{f}$$ is also real-valued (see also Exercise 4.1.4(b)). Therefore the polynomial $$f + i\widetilde{f}$$ contains only positive frequencies. Thus for $$k \in \mathbf{Z}^{+}$$ we have

Expanding the 2k power and taking real parts, we obtain

where we used that f is real-valued. Therefore,

by applying Hölder’s inequality with exponents 2k∕(2k − 2j) and 2k∕(2j) to the jth term of the sum. Dividing the last inequality by $$\|f\|_{L^{2k}}^{2k}$$, we obtain

(4.1.13)

where $$R =\|\widetilde{ f}\,\|_{L^{2k}}/\|f\|_{L^{2k}}$$. If R > 0 satisfies (4.1.13), then $$R \leq C_{2k}$$, where C 2k is the largest real root of the polynomial $$g(t) = t^{2k} -\sum _{j=1}^{k}\binom{2k}{2j}t^{2k-2j}$$. (Since g(0) < 0 and $$\lim _{t\rightarrow \infty }g(t) = \infty $$, g has at least one real root.) We conclude that if f satisfies (a), (b), and (c), then we have for all $$k = 1,2,\ldots$$

(4.1.14)

We now remove assumptions (a), (b), and (c). We first remove assumption (c). Given a complex-valued trigonometric polynomial f with $$\widehat{f}(0) = 0$$, we write

(with c ₀ = 0) and we note that the expressions inside the square brackets are real-valued trigonometric polynomials. Thus we can express f as P + iQ, where P and Q are real-valued trigonometric polynomials, and applying (4.1.14) to P and Q we obtain the inequality

(4.1.15)

for all trigonometric polynomials f with $$\widehat{f}(0) = 0$$.

Next, we remove the assumption that $$\widehat{f}(0) = 0$$. We write $$f = (f -\widehat{ f}(0)) +\widehat{ f}(0)$$, we observe that the conjugate function of a constant is zero, and we apply (4.1.15) to obtain

Since trigonometric polynomials are dense in $$L^{p}$$, it follows that the operator $$f\mapsto \widetilde{f}$$ has a bounded extension on L 2k that satisfies (4.1.12) for all f ∈ L 2k , and in particular for all $$f \in \mathcal{C}^{\infty }(\mathbf{T}^{1})$$.

Every real number p ≥ 2 lies in an interval of the form [2k, 2k + 2], for some $$k \in \mathbf{Z}^{+}$$. Theorem 1.​3.​4 gives that for all $$2 \leq p < \infty $$ there is a constant A p such that

(4.1.16)

when f is a simple function. Thus the conjugate function has a bounded extension on $$L^{p}$$ that satisfies (4.1.16) when p ≥ 2.

To extend this result for p < 2 we use duality. We observe that the adjoint operator of $$f\mapsto \widetilde{f}$$ is $$f\mapsto -\widetilde{ f}$$. Indeed, for f, g in $$\mathcal{C}^{\infty }(\mathbf{T}^{1})$$ we have

By duality, estimate (4.1.16) is also valid for 1 < p ≤ 2 with constant $$A_{p'} = A_{p}$$. □ 

We extend the preceding result to higher dimensions.

Theorem 4.1.8.

Let 1 < p < ∞ and $$f \in L^{p}(\mathbf{T}^{n})$$ . Then $$D_{N}^{n} {\ast} f$$ converges to f in $$L^{p}$$ as $$N \rightarrow \infty $$ .

Proof.

As a consequence of Corollary 4.1.3, Proposition 4.1.6, and Theorem 4.1.7, it suffices to show that for all f trigonometric polynomials on T n we have

Obviously, this inequality is valid in dimension n = 1. We extend it by induction to all dimensions. We assume that it is valid in dimension n − 1 and we prove it in dimension n.

Let $$x' = (x_{2},\ldots,x_{n}) \in \mathbf{T}^{n-1}$$. For a fixed trigonometric polynomial f, and for fixed N ≥ 0 and x′ ∈ T n−1, define a trigonometric polynomial g N, x′ on $$\mathbf{T}^{1}$$ by setting

where $$m' = (m_{2},\ldots,m_{n})$$. Then we have

where $$f_{x_{1}}$$ is the trigonometric polynomial of $$n - 1$$ variables defined by $$f_{x_{1}}(x') = f(x_{1},x')$$. We also have that

Combining this information, we write

where the penultimate inequality follows from the induction hypothesis. □ 

4.1.3 Bochner–Riesz Summability

In dimension 1 the Fejér means of an integrable function are better behaved than the Dirichlet means. We investigate whether there is a similar phenomenon in higher dimensions. Recall that the circular (or spherical) partial sums of the Fourier series of f are given by

where R ≥ 0. Taking the averages of these expressions, we obtain

and we call these expressions the circular Cesàro means (or circular Fejér means) of f. It turns out that the circular Cesàro means of integrable functions on T ² always converge in L ¹, but in dimension 3, this may fail. Theorem 4.2.5 gives an example of an integrable function f on T ³ whose circular Cesàro means diverge a.e. However, we show below that this is not the case if the circular Cesàro means of a function f in $$L^{1}(\mathbf{T}^{3})$$ are replaced by the only slightly different-looking means

for some $$\varepsilon > 0$$. This discussion suggests that the preceding expressions behave better as $$\varepsilon$$ increases, but for a fixed $$\varepsilon$$ they get worse as the dimension increases. The need to understand the behavior of these operators for different values of $$\alpha \geq 0$$ led to introduction of the operators $$B_{R}^{\alpha }$$ given in Definition 4.1.2.

The family of operators $$B_{R}^{\alpha }$$ forms a natural “spherical” analogue of the Cesàro–Fejér sums. It turns out that there is no significant difference in the behavior of these means if the expression $$\big(1 -\frac{\vert m\vert ^{2}} {R^{2}} \big)^{\alpha }$$ in (4.1.6) is replaced by the expression $$\big(1 -\frac{\vert m\vert } {R} \big)^{\alpha }$$; see Exercise 4.3.1. The advantage of the quadratic expression in (4.1.6) is that it has an easily computable kernel and yields the elegant reproducing formula

(4.1.17)

which precisely quantifies the way in which $$B_{R}^{\alpha }$$ is smoother than $$B_{R}^{\beta }$$ when $$\alpha >\beta$$. Identity (4.1.17) also says that when $$\alpha >\beta$$, the operator $$B_{R}^{\alpha }(f)$$ is an average of the operators $$B_{r}^{\beta }(f)$$, 0 < r < R, with respect to a certain density.

Note that the Bochner–Riesz means of order zero coincide with the circular (or spherical) Dirichlet means, and, as we have seen, these converge in L ²(T n ). We address an analogous question on $$L^{p}(\mathbf{T}^{n})$$ for p ≠ 2.

Proposition 4.1.9.

Let $$1 \leq p < \infty $$ and $$f \in L^{p}(\mathbf{T}^{n})$$ . Then the Bochner–Riesz means $$B_{R}^{\alpha }(f)$$ converge to f in $$L^{p}(\mathbf{T}^{n})$$ as $$R \rightarrow \infty $$ when $$\alpha > (n - 1)\big\vert \tfrac{1} {p} -\tfrac{1} {2}\big\vert $$ . Moreover, if f is continuous on $$\mathbf{T}^{n}$$ and $$\alpha > \tfrac{n-1} {2}$$ , then $$B_{R}^{\alpha }(f)$$ converges to f uniformly as $$R \rightarrow \infty $$ .

Proof.

For $$z \in \mathbf{C}$$ with $$\mbox{ Re}\,z \geq 0$$, consider the function

defined for ξ in $$\mathbf{R}^{n}$$. Note that $$\|m_{z}\|_{L^{\infty }} = 1$$. Using an identity proved in Appendix B.5, we have that

(4.1.18)

where $$y \in \mathbf{R}^{n}$$ and J ν is the Bessel function of order ν. The estimates in Appendices B.6 and B.7 imply that there is a constant $$C(\mbox{ Re}\,\nu )$$ such that

whenever $$\mbox{ Re}\,\nu > 0$$. This yields that if $$\mbox{ Re}\,z > \frac{n-1} {2}$$, then there is a constant C′(t) such that the function K z obeys the estimate

(4.1.19)

and hence it lies in $$L^{1}(\mathbf{R}^{n})$$. Using identity (3.​1.​10), whenever $$\mbox{ Re}\,z > \frac{n-1} {2}$$, we define for an integrable function f on $$\mathbf{T}^{n}$$ and $$x \in \mathbf{T}^{n}$$ the operator

where L z, R is a function whose sequence of Fourier coefficients is $$\{m_{z}( \tfrac{\ell}{R})\}_{\ell\in \mathbf{Z}^{n}}$$. But the function L z, R can be precisely identified. By the Poisson summation formula (Theorem 3.​2.​8), which applies since both K z (x) and m z (x) are bounded by a constant multiple of $$(1 + \vert x\vert )^{-n-\delta }$$ for some $$\delta > 0$$, we have

for all $$x \in \mathbf{T}^{n}$$. We show that the family $$\{L^{z,R}\}_{R>0}$$ is an approximate identity on $$\mathbf{T}^{n}$$ when $$\mbox{ Re}\,z > \frac{n-1} {2}$$; on this see the related Exercise 3.1.3. Obviously, using (4.1.19) we have that

(4.1.20)

for some constant $$C''(n,\mbox{ Re}\,z)$$, and also

for all R > 0 when $$\mbox{ Re}\,z > \frac{n-1} {2}$$. Moreover, for $$\delta < \frac{1} {2}$$ using (4.1.19) we have

thus the integral of L z, R over $$[-1/2,1/2]^{n}\setminus [-\delta,\delta ]^{n}$$ tends to zero as $$R \rightarrow \infty $$.

Using Theorem 1.​2.​19, we obtain these conclusions for $$\mbox{ Re}\,z > \frac{n-1} {2}$$:

(a)

For $$f \in L^{1}(\mathbf{T}^{n})$$, $$B_{R}^{z}(f)$$ converge to f in L ¹ as $$R \rightarrow \infty $$.

(b)

For f continuous on $$\mathbf{T}^{n}$$, $$B_{R}^{z}(f)$$ converge to f uniformly as $$R \rightarrow \infty $$.

We turn to the corresponding results for $$1 < p < \infty $$. We have that

(4.1.21)

(4.1.22)

The family of operators $$f\mapsto B_{R}^{z}(f)$$ is of admissible growth for all $$\mbox{ Re}\,z \geq 0$$, since for all measurable subsets A, B of $$\mathbf{T}^{n}$$ we have

thus condition (1.​3.​23) holds. Moreover, hypothesis (1.​3.​24) of Theorem 1.​3.​7 holds in view of (4.1.21) and (4.1.22). Applying Theorem 1.​3.​7 (or rather Exercise 1.3.4 in which the strip $$[0,1] \times \mathbf{R}$$ is replaced by the more general strip $$[a,b] \times \mathbf{R}$$) we obtain that when $$\alpha = \mbox{ Re}\,z > (n - 1)\vert \frac{1} {p} -\frac{1} {2}\vert $$, we have

Finally, using Corollary 4.1.3, we deduce that $$B_{R}^{\alpha }(f) \rightarrow f$$ in $$L^{p}(\mathbf{T}^{n})$$ as $$R \rightarrow \infty $$ for all $$f \in L^{p}(\mathbf{T}^{n})$$. □ 

The preceding result is sharp in the case p = 1 (Theorem 4.2.5). For this reason, the number $$\alpha = (n - 1)/2$$ is referred to as the critical index of Bochner–Riesz summability .

Exercises

4.1.1. If $$f \in \mathcal{C}^{\infty }(\mathbf{T}^{n})$$, then show that $$D_{N}^{n} {\ast} f$$ and $$ D_{N }^{n } {\ast}f$$ converge to f uniformly and in $$L^{p}$$ for $$1 \leq p \leq \infty $$.

4.1.2. Prove that

where $$W(f) =\widetilde{ f}$$ is the conjugate function on the circle. Moreover, show that the mappings $$f\mapsto W(f) +\widehat{ f}(0)$$ and $$f\mapsto W(f) -\widehat{ f}(0)$$ are isometries on $$L^{2}(\mathbf{T}^{1})$$.

4.1.3. Let $$-\infty \leq a_{j} < b_{j} \leq +\infty $$ for $$1 \leq j \leq n$$. Consider the rectangular projection operator defined on $$\mathcal{C}^{\infty }(\mathbf{T}^{n})$$ by

Prove that when $$1 < p < \infty $$, P extends to a bounded operator from $$L^{p}(\mathbf{T}^{n})$$ to itself with bounds independent of the $$a_{j},b_{j}$$.

[Hint: Express P in terms of the Riesz projection P ₊.]

4.1.4. Let P r (t) be the Poisson kernel on $$\mathbf{T}^{1}$$ as defined in Exercise 3.1.7. For 0 < r < 1, define the conjugate Poisson kernel Q r (t) on the circle by

(a) For 0 < r < 1, prove the identity

(b) Prove that $$\widetilde{f}(t) =\lim _{r\rightarrow 1}(Q_{r} {\ast} f)(t)$$ whenever f is smooth. Conclude that if f is real-valued, then so is $$\widetilde{f}$$.

(c) Let $$f \in L^{1}(\mathbf{T}^{1})$$. Prove that the function

is analytic in $$z = re^{2\pi it}$$ on the open unit disc $$\{z \in \mathbf{C}:\,\, \vert z\vert < 1\}$$.

(d) Let $$f \in L^{1}(\mathbf{T}^{1})$$. Conclude that the functions $$z\mapsto (P_{r} {\ast} f)(t)$$ and $$z\mapsto (Q_{r} {\ast} f)(t)$$ are conjugate harmonic functions of $$z = re^{2\pi it}$$ in the region | z |  < 1. The term conjugate Poisson kernel stems from this property.

4.1.5. Let f be in $$\dot{\varLambda }_{\alpha }(\mathbf{T}^{1})$$ for some $$0 <\alpha < 1$$. Prove that the conjugate function $$\widetilde{f}$$ is well defined and can be written as

[Hint: Use part (b) of Exercise 4.1.4 and the fact that Q r has integral zero over the circle to write $$(f {\ast} Q_{r})(x) =\big ((f - f(x)) {\ast} Q_{r}\big)(x)$$, allowing use of the Lebesgue dominated convergence theorem.]

4.1.6. Suppose that f is a real-valued function on $$\mathbf{T}^{1}$$ with | f | ≤ 1 and $$0 \leq \lambda <\pi /2$$.

(a) Prove that

(b) Conclude that for $$0 \leq \lambda <\pi /2$$ we have

[Hint: Part (a): Consider the analytic function F(z) on the disk | z |  < 1 defined by $$F(z) = -i(P_{r} {\ast} f)(\theta ) + (Q_{r} {\ast} f)(\theta )$$, where $$z = re^{2\pi i\theta }$$. Then $$\mbox{ Re }e^{\lambda F(z)}$$ is harmonic and its average over the circle | z |  = r is equal to its value at the origin, which is $$\cos (\lambda f(0)) \leq 1$$. Let $$r \uparrow 1$$ and use that for z = e 2π i t on the circle we have $$\mbox{ Re }e^{\lambda F(z)} \geq e^{\lambda \widetilde{f}(t)}\cos (\lambda )$$.]

4.1.7. Prove that for $$0 <\alpha < 1$$ there is a constant $$C_{\alpha }$$ such that

[Hint: Using Exercise 4.1.5, for $$\vert h\vert \leq 1/10$$ write $$\widetilde{f}(x + h) -\widetilde{ f}(x)$$ as

You may use the fact that $$\cot (\pi t) = \frac{1} {\pi t} + b(t)$$, where b(t) is a bounded function when | t | ≤ 1∕2. The case | h | ≥ 1∕10 is easy.]

4.1.8. The beta function is defined in Appendix A.2. Derive the identity

and show that the function $$K_{R}^{\alpha }(x) =\sum _{\vert m\vert \leq R}\big(1 -\frac{\vert m\vert ^{2}} {R^{2}} \big)^{\alpha }e^{2\pi im\cdot x}$$ satisfies (4.1.17).

[Hint: Take $$t = 1 -\frac{\vert m\vert ^{2}} {R^{2}}$$ and change variables $$s = \tfrac{r^{2}-\vert m\vert ^{2}} {R^{2}}$$ in the displayed identity.]

4.2 A. E. Divergence of Fourier Series and Bochner–Riesz means

We saw in Proposition 3.​4.​6 that the Fourier series of a continuous function may diverge at a point. As expected, the situation can only get worse as the functions get worse. In this section we present an example, due to A. N. Kolmogorov, of an integrable function on $$\mathbf{T}^{1}$$ whose Fourier series diverges almost everywhere. We also prove an analogous result for the Bochner–Riesz means at the critical index.

4.2.1 Divergence of Fourier Series of Integrable Functions

It is natural to start our investigation with the case n = 1. We begin with the following important result:

Theorem 4.2.1.

There exists an integrable function on the circle $$\mathbf{T}^{1}$$ whose Fourier series diverges almost everywhere.

Proof.

The proof of this theorem is a bit involved, and we need a sequence of lemmas, which we prove first.

Lemma 4.2.2.

(Kronecker) Suppose that $$N \in \mathbf{Z}^{+}$$ and

is a linearly independent set over the rationals. Then for any $$\varepsilon > 0$$ and any complex numbers $$z_{1},z_{2},\ldots,z_{N}$$ with |z j | = 1, there exists an integer $$L \in \mathbf{Z}$$ such that

Proof.

Suppose that the assertion claimed is false. Then there is an $$\varepsilon > 0$$ and complex numbers $$z_{j} = e^{2\pi i\theta _{j}}$$, $$j = 1,\ldots,N$$, with 0 ≤ θ j  < 1, such that

where $$B\big((\theta _{1},\ldots,\theta _{N}),\varepsilon \big)$$ denotes a neighborhood in $$\mathbf{T}^{N}$$ of radius $$\varepsilon$$ centered at the point $$(\theta _{1},\ldots,\theta _{N})$$. Pick a smooth, nonzero, and nonnegative function f on T N supported in $$B\big((\theta _{1},\ldots,\theta _{N}),\varepsilon \big)$$. Then $$f(m(x_{1},\ldots,x_{N})) = 0$$ for all $$m \in \mathbf{Z}$$, but

(4.2.1)

Set $$x = (x_{1},\ldots,x_{N})$$. Then we have

Note that $$e^{2\pi i(\ell\cdot x)} - 1\neq 0$$ because $$\ell\cdot x =\ell _{1}x_{1} + \cdots +\ell _{N}x_{N}\notin \mathbf{Z}$$, since by assumption the set $$\{x_{1},x_{2},\ldots,x_{N},1\}$$ is linearly independent over the rationals. Observe that

tends to 0 as $$M \rightarrow \infty $$ for every fixed $$\ell\in \mathbf{Z}^{N}$$ and is bounded uniformly in M by $$\vert \widehat{f}(\ell)\vert $$ which satisfies $$\sum _{\ell\in \mathbf{Z}^{N}}\vert \widehat{f}(\ell)\vert < \infty $$. Using the Lebesgue dominated convergence theorem, we obtain that

which contradicts (4.2.1). Therefore the claimed L exists. □ 

Lemma 4.2.3.

There exists a positive constant c > 0 such that given any integer N ≥ 2 there exists a positive measure μ N on $$\mathbf{T}^{1}$$ with $$\mu _{N}(\mathbf{T}^{1}) = 1$$ such that

(4.2.2)

for almost all $$x \in \mathbf{T}^{1}$$ (c is a fixed constant).

Proof.

Given irrational real numbers $$x_{1},\ldots,x_{N}$$ such that the set $$\{x_{1},\ldots,x_{N},1\}$$ is linearly independent over the rationals, we define $$\mathbf{Q}[x_{1},\ldots,x_{N}]$$ to be the field extension of Q consisting of all linear combinations of the form $$q_{0} + q_{1}x_{1} + \cdots + q_{N}x_{N}$$, where q j are rational numbers. Obviously $$\mathbf{Q}[x_{1},\ldots,x_{N}]$$ is a countable set. Fix N ≥ 100 and choose points x j as follows:

(4.2.3)

and such that $$x_{1}\notin \mathbf{Q}$$, $$x_{2}\notin \mathbf{Q}[x_{1}]$$, $$\ldots$$, $$x_{N}\notin \mathbf{Q}[x_{1},\ldots,x_{N-1}]$$. Then obviously the set $$\{x_{1},\ldots,x_{N},1\}$$ is linearly independent over the rationals. Let

and observe that every x in $$[0,1]\setminus \mathbf{Q}[x_{1},\ldots,x_{N}]$$ belongs to E N . Indeed, if $$x\notin E_{N}$$, then there are rational numbers q j such that

Then $$q = q_{1} + \cdots + q_{N}\neq 0$$, since $$\{x_{1},\ldots,x_{N},1\}$$ are linearly independent over Q. It follows that

thus $$x \in \mathbf{Q}[x_{1},\ldots,x_{N}]$$. We conclude that E N has full measure.

Next, we define the probability measure

where $$\delta _{x_{j}}$$ are Dirac delta masses at the points x j . For this measure we have

(4.2.4)

where the signum function is defined as sgn a = 1 for a > 0, − 1 for a < 0, and zero if a = 0. By Lemma 4.2.2, for all x ∈ E N there exists an $$L \in \mathbf{Z}^{+}$$ such that

which can be equivalently written as

(4.2.5)

It follows from (4.2.5) that

Combining this with the result of the calculation in (4.2.4), we obtain that

But for every x ∈ [0, 1), there exists a j ₀ such that $$x \in [x_{j_{0}},x_{j_{0}+1})$$. It follows from (4.2.3) that $$\vert x - x_{j}\vert \leq C(\vert j - j_{0}\vert + 1)N^{-1}$$, and thus

Thus for every x ∈ E N there exists an $$L \in \mathbf{Z}^{+}$$ such that

which proves the required conclusion since E N is a set of measure 1. □ 

Lemma 4.2.4.

For each $$0 < M < \infty $$ there exists a trigonometric polynomial g M and a measurable subset A M of $$\mathbf{T}^{1}$$ with measure $$\vert A_{M}\vert > 1 - 2^{-M}$$ such that $$\|g_{M}\|_{L^{1}} = 1$$ , and such that

(4.2.6)

Proof.

Given an $$M \in \mathbf{Z}^{+}$$, we pick an integer N(M) such that clogN(M) > 2 M+2, where c is as in (4.2.2), and we also pick the measure μ N(M), which satisfies (4.2.2). By Fatou’s lemma we have

and thus we can find a positive integer L(M) such that the set

has measure greater than 1 − 2−M . We pick a positive integer K(M) such that

where F K is the Fejér kernel. This is possible, since the Fejér kernel is an approximate identity and $$\{D_{j}:\,\, 1 \leq j \leq L(M)\}$$ is a finite family of continuous functions. Then we define $$g_{M} =\mu _{N(M)} {\ast} F_{K(M)}$$. Since μ N(M) is a probability measure, we obtain

for all x ∈ [0, 1] and 1 ≤ j ≤ L(M). But given x ∈ A M there exists an L in $$\{1,\ldots,L(M)\}$$ such that $$\vert (D_{L} {\ast}\mu _{N(M)})(x)\vert \geq 2^{M+2}$$ and for this L we have

Therefore, (4.2.6) is satisfied for this g M and A M . Since μ N is a nonnegative measure and F K(M) is nonnegative and has L ¹ norm 1, we have that

showing that g M has L ¹ norm equal to one. □ 

We now have the tools needed to construct an example of a function whose Fourier series diverges almost everywhere. The example is given as a series of functions whose behavior worsens as its index becomes bigger. The function we wish to construct is a sum of the form

(4.2.7)

for a choice of sequences $$\varepsilon _{j} \rightarrow 0$$ and $$M_{j} \rightarrow \infty $$, where g M are as in Lemma 4.2.4.

Let us be specific. First, we set d ₀ = 1 and for N ≥ 1

(4.2.8)

where g M is the trigonometric polynomial of Lemma 4.2.4. We set $$\varepsilon _{0} = M_{0} = 1$$. Assume that we have defined $$\varepsilon _{j}$$ and $$M_{j}$$ for all 1 ≤ j < N for some N ≥ 2. We set

(4.2.9)

and then we pick M N such that

(4.2.10)

This defines $$\varepsilon _{N}$$ and M N for a given positive integer N, provided $$\varepsilon _{j}$$ and M j are known for all j < N. This way we define $$\varepsilon _{N}$$ and M N for all natural numbers N.

We observe that the selections of $$\varepsilon _{j}$$ and M j force the inequalities $$\varepsilon _{j} \leq 2^{-j}$$ and $$d_{j} \leq d_{j+1}$$ for all j ≥ 1. Since each $$g_{M_{j}}$$ has L ¹ norm 1 and $$\varepsilon _{j} \leq 2^{-j}$$, the function g in (4.2.7) is integrable and has L ¹ norm at most 1.

For a given j ≥ 1 and $$x \in A_{M_{j}}$$, by Lemma 4.2.4 there exists an L ≥ 1 such that $$\vert (D_{L} {\ast} g_{M_{j}})(x)\vert > 2^{M_{j}}$$. Set

Then we have

We make the following observations:

(i)

$$\vert (D_{k} {\ast} g_{M_{j}})(x)\vert = \vert (D_{L} {\ast} g_{M_{j}})(x)\vert > 2^{M_{j}}$$.

(ii)

$$\vert (D_{k} {\ast} g_{M_{s}})(x)\vert = \vert (D_{\min (d_{s},k)} {\ast} g_{M_{s}})(x)\vert \leq \big\| D_{\min (d_{s},L)}\big\|_{L^{\infty }} \leq 3d_{s}$$, when $$s < j$$.

(iii)

$$\vert (D_{k} {\ast} g_{M_{s}})(x)\vert = \vert (D_{\min (d_{s},k)} {\ast} g_{M_{s}})(x)\vert \leq \big\| D_{\min (d_{j},L)}\big\|_{L^{\infty }} \leq 3d_{j}$$, when s > j.

In these estimates we have used that k = min(L, d j ), $$\big\|D_{m}\big\|_{L^{\infty }} \leq 2m + 1 \leq 3m$$, and that

which follows easily by examining the corresponding Fourier coefficients.

Using the estimates in (i), (ii), and (iii), for a fixed $$x \in A_{M_{j}}$$ and k = k(x) we obtain

(4.2.11)

Our selection of $$\varepsilon _{j}$$ and M j now ensures that (4.2.11) is a large number. In fact, we have

and

Therefore, the expression in (4.2.11) is at least $$\varepsilon _{j}2^{M_{j}} - d_{j-1} - 1 \geq 2^{j}$$. It follows that for every j ≥ 1 and every $$x \in A_{M_{j}}$$ there exists a $$k = k(x) \in \mathbf{Z}^{+}$$ such that

We conclude that for every r ≥ 1 and $$x \in \bigcup \limits _{j=r}^{\infty }A_{M_{j}}$$,

(4.2.12)

since x belongs to some $$A_{M_{j}}$$ with $$j \geq r$$. For given r ≥ 1, Lemma 4.2.4 yields that

Then the set

has measure 1, since it is a countable intersection of subsets of $$\mathbf{T}^{1}$$ of full measure. In view of (4.2.12) we have that for all x in A

(4.2.13)

and thus the required conclusion follows. □ 

4.2.2 Divergence of Bochner–Riesz Means of Integrable Functions

We now turn to the corresponding n-dimensional problem for spherical summability of Fourier series. The situation here is quite similar at the critical index $$\alpha = \tfrac{n-1} {2}$$.

Theorem 4.2.5.

Let n > 1. There exists an integrable function f on $$\mathbf{T}^{n}$$ such that

for almost all $$x \in \mathbf{T}^{n}$$ . Furthermore, such a function can be constructed such that it is supported in an arbitrarily small given neighborhood of the origin.

We will need a couple of lemmas.

Lemma 4.2.6.

Let n ≥ 2. The complement of the set

has n-dimensional Lebesgue measure zero.

Proof.

Recall that a function g defined on an open subset $$\varOmega$$ of $$\mathbf{R}^{n}$$ is called real analytic if for every point x ₀ in $$\varOmega$$ there is a ball $$B(x_{0},\varepsilon )$$ is contained in $$\varOmega$$ and there exist coefficients $$c_{\beta }(x_{0})$$ such that $$g(x) =\sum _{\beta }c_{\beta }(x_{0})(x - x_{0})^{\beta }$$ for all $$\vert x - x_{0}\vert <\varepsilon$$, where the sum is taken over all multiindices. We will need two facts about real analytic functions. First, the function x → | x | is real analytic on $$\mathbf{R}^{n}\setminus \{0\}$$. Indeed, given x ₀ ≠ 0, for $$\vert x - x_{0}\vert < \vert x_{0}\vert /3$$ we have that

This allows us to write

which is a power series of the form $$\sum _{\beta }c_{\beta }(x_{0})(x - x_{0})^{\beta }$$.

Secondly, we need the fact that a real analytic function defined on an open connected subset of $$\mathbf{R}^{n}$$ cannot vanish on a set of positive measure, unless it is identically equal to zero; a proof of this in dimension one and an outline of the proof in higher dimensions is contained in [205].

We return to the proof of the lemma which requires us to show that S has full measure in $$\mathbf{R}^{n}$$. Indeed, if $$x \in \mathbf{R}^{n}\setminus S$$, then there exist $$k \in \mathbf{Z}^{+}$$, $$m_{1},\ldots m_{k} \in \mathbf{Z}^{n}$$, and $$q_{0},q_{1},\ldots,q_{k}$$ nonzero rational numbers such that

(4.2.14)

Since the function

is nonzero and real analytic on $$\mathbf{R}^{n}\setminus \mathbf{Z}^{n}$$, it must vanish only on a set of Lebesgue measure zero. Therefore, there exists a set $$A_{m_{1},\ldots,m_{k},q_{0},q_{1},\ldots,q_{k}}$$ of Lebesgue measure zero such that (4.2.14) holds exactly when x is in this set. Then

from which it follows that $$\mathbf{R}^{n}\setminus S$$ has Lebesgue measure zero. □ 

Let us denote the Bochner–Riesz kernel by

when $$x \in \mathbf{T}^{n}$$. We need the following lemma regarding $$K_{R}^{\alpha }$$:

Lemma 4.2.7.

Let n ≥ 2. For almost every $$x \in \mathbf{T}^{n}$$ we have

It is noteworthy to compare the result of this lemma with the analogous one-dimensional statement

for the Dirichlet kernel, which holds exactly when x = 0. Thus the uniform ill behavior of the kernel $$K_{R}^{\frac{n-1} {2} }$$ reflects in some sense its lack of localization.

Proof.

Fix n ≥ 2 and fix $$x_{0} \in \big ([-1/2,1/2)^{n}\setminus \{0\}\big) \cap S$$, where S is as in Lemma 4.2.6. Using (4.1.18) and the Poisson summation formula (Theorem 3.​2.​8), for each $$\alpha > \tfrac{n-1} {2}$$ we obtain the identity

(4.2.15)

and the sum converges absolutely because of the asymptotics for the Bessel functions in Appendix B.8. The term with m = 0 in the sum in (4.2.15) is a finite constant since by Appendix B.6 the function

is smooth and therefore bounded. But for $$m\neq 0$$ in (4.2.15) we have | m − x ₀ | ≥ 1∕2. The asymptotics in Appendix B.8 imply that for R ≥ 2 we have

for all $$\alpha > \tfrac{n-1} {2}$$. We insert this expression in (4.2.15), we multiply by $$e^{2\pi i\lambda R}$$ for some $$\lambda$$ real, and then we average in R from 1 to T, for some T > 10. We obtain

(4.2.16)

Assume that we are able to pass the limit as $$\alpha \rightarrow \tfrac{n-1} {2} +$$ through the sums and integrals in the preceding identity; we justify this step momentarily. Then we obtain

(4.2.17)

We now justify the passage of the limit in $$\alpha$$ inside the sums and the integrals in (4.2.16) to obtain (4.2.17). First, when $$\vert m\vert \leq R \leq T$$ and $$\alpha > \frac{n-1} {2}$$, the mean value theorem gives

thus $$K_{R}^{\alpha }(x_{0})$$ converges to $$K_{R}^{\frac{n-1} {2} }(x_{0})$$ uniformly in $$R \in [1,T]$$ as $$\alpha \rightarrow \frac{n-1} {2}$$ and therefore the integral over [1, T] of the former converges to the integral over [1, T] of the latter. Next, an integration by parts shows that the integral

is bounded by a constant multiple of $$(\lambda +\vert x_{0} - m\vert )^{-1}$$, which makes the first infinite sum in (4.2.16) converge absolutely and uniformly in $$\alpha \geq \frac{n-1} {2}$$, thus one may pass the limit in $$\alpha$$ inside the sum. Also, the integral

is bounded by a constant multiple of $$(\lambda -\vert x_{0} - m\vert )^{-1}$$ whenever $$\lambda$$ is not in the set

where $$0 <\lambda _{1} <\lambda _{2} <\lambda _{3} < \cdots $$. Thus for $$\lambda \notin \varLambda _{x_{0}}$$, the preceding argument explains the passage of the limit in $$\alpha$$ inside the second infinite sum in (4.2.16). If $$\lambda$$ happens to be in $$\varLambda _{x_{0}}$$, then there is at most one m ₀ ≠ 0, such that $$\lambda = \vert x_{0} - m_{0}\vert $$ and the second sum in (4.2.16) restricted to $$m \in \mathbf{Z}^{n}\setminus \{0,m_{0}\}$$ converges absolutely, while for the single term with m = m ₀, letting $$\alpha \rightarrow \frac{n-1} {2} +$$ is trivial. Finally, for the term involving the Bessel function $$ J_{n- \frac{1}{2}}$$, the passage of the limit in $$\alpha$$ inside the integral is straightforward since the function

is continuous on the compact set $$[\frac{n-1} {2}, \frac{n} {2} ] \times [1,T]$$. This completes the proof of (4.2.17).

There are four terms to the right of (4.2.17) and we observe that if $$\lambda \neq \pm \vert x_{0} - m_{0}\vert $$ for any $$m_{0} \in \mathbf{Z}^{n}$$, then all these terms converge to zero as $$T \rightarrow \infty $$. This assertion is trivial for the first three of these four terms, while for the last we assume that $$T > \vert x_{0}\vert ^{-1}$$. We split the integral

(4.2.18)

as a sum of the integral over $$[1,\vert x_{0}\vert ^{-1}]$$, which obviously converges to zero as $$T \rightarrow \infty $$ by Appendix B.6, and of the integral over $$[\,\vert x_{0}\vert ^{-1},T]$$. For the latter, we use the asymptotics in Appendix B.8 to write

The part of the integral in (4.2.18) over $$[\,\vert x_{0}\vert ^{-1},T]$$ corresponding to $$O\big((R\vert x_{0}\vert )^{-\frac{3} {2} }\big)$$ grows like $$\log (T\vert x_{0}\vert )$$ which divided by $$T$$ obviously tends to zero. The part of the integral in (4.2.18) over $$[\,\vert x_{0}\vert ^{-1},T]$$ corresponding to the main term is

which tends to zero as $$T \rightarrow \infty $$ by an integration by parts, since $$\lambda \neq \vert x_{0}\vert $$ because we are considering the case where $$\lambda \neq \pm \vert x_{0} - m\vert $$ for any $$m \in \mathbf{Z}^{n}$$.

Now consider the case where $$\lambda = \pm \vert x_{0} - m_{0}\vert $$ for some $$m_{0} \in \mathbf{Z}^{n}$$. In this case the expression to the right in (4.2.17) converges to

as $$T \rightarrow \infty $$. Next observe that

(4.2.19)

We have now shown that

(4.2.20)

Since x ₀ lies in S, the set $$\{1\} \cup \{\lambda _{1},\lambda _{2},\lambda _{3},\ldots \}$$ is linearly independent over the rationals and thus no expression of the form $$\pm \lambda _{j_{1}} \pm \ldots \pm \lambda _{j_{s}}$$ is equal to an integer. It follows from this fact and (4.2.20) that

Suppose we had that

Then, setting $$c_{n} = \frac{\varGamma (\tfrac{n+1} {2} )} {\pi ^{\frac{n-1} {2} }}$$, we would have

which contradicts (4.2.19) by letting $$N \rightarrow \infty $$. Here, once again, we used the fact that no expression of the form $$\pm \lambda _{j_{1}} \pm \ldots \pm \lambda _{j_{s}}$$ is equal to an integer and thus the preceding $$\limsup$$ is a limit and is equal to 1, since the integral of all the exponentials produces another exponential which remains bounded.

We deduce that $$\sup _{R\geq 1}\vert K_{R}^{\frac{n-1} {2} }(x_{0})\vert = \infty $$ for every point $$x_{0} \in S \cap [-\frac{1} {2}, \frac{1} {2})^{n}\setminus \{0\}$$ and this concludes the proof of Lemma 4.2.7. □ 

Proof.

We now prove Theorem 4.2.5. This part of the proof is similar to the proof of Theorem 4.2.1. Lemma 4.2.7 says that the means $$B_{R}^{\frac{n-1} {2} }(\delta _{0})(x)$$, where $$\delta _{0}$$ is the Dirac mass at 0, do not converge for almost all $$x \in \mathbf{T}^{n}$$. Our goal is to replace this Dirac mass by a series of integrable functions on T n that have a peak at the origin.

Let us fix a nonnegative $$\mathcal{C}^{\infty }$$ radial function $$\widehat{\varPhi }$$ on $$\mathbf{R}^{n}$$ that is supported in the unit ball | ξ | ≤ 1 and has integral equal to 1. We set

where the identity is valid because of the Poisson summation formula. It follows that the mth Fourier coefficient of $$\varphi _{\varepsilon }$$ is $$\varPhi (\varepsilon m)$$. Therefore, we have the estimate

(4.2.21)

For any k ≥ 1, we construct measurable subsets E k of $$\mathbf{T}^{n}$$ with $$\vert E_{k}\vert \geq 1 -\tfrac{1} {k}$$, a sequence of positive numbers $$R_{1} < R_{2} < \cdots $$, with $$R_{k} \uparrow \infty $$, and two sequences of positive numbers $$\varepsilon _{k} \downarrow 0$$ and $$\gamma _{k} \downarrow 0$$ such that $$\varepsilon _{k} \leq \gamma _{k}$$ for all k and

(4.2.22)

We pick E ₁ = ∅, R ₁ = 1, and $$\varepsilon _{1} =\gamma _{1} = 1$$. Let k > 1 and suppose that we have selected E j , R j , $$\gamma _{j}$$, and $$\varepsilon _{j}$$ for all 1 ≤ j ≤ k − 1 such that (4.2.22) is satisfied. We construct E k , R k , $$\gamma _{k}$$, and $$\varepsilon _{k}$$ such that (4.2.22) is satisfied with j = k. We begin by choosing $$\gamma _{k}$$. Let B be a constant such that

for all $$x,y \in \mathbf{R}^{n}$$. Define $$\gamma _{k}$$ such that

(4.2.23)

Then define

where C n is the constant in (4.2.21), and observe that in view of (4.2.21) we have

(4.2.24)

Let $$\delta _{0}$$ be the Dirac mass at the origin in $$\mathbf{T}^{n}$$. Since by Fatou’s lemma and Lemma 4.2.7 we have

there exists an $$R_{k} >\max (R_{k-1},k)$$ such that the set

has measure at least $$1 -\tfrac{1} {k}$$. Note that since R k is increasing and tends to infinity, (4.2.23) yields that $$\gamma _{k}$$ is decreasing and tends to zero.

We now choose $$\varepsilon _{k}$$ such that $$\varepsilon _{k} \leq \gamma _{k}$$, $$\varepsilon _{k} \leq \varepsilon _{k-1}$$, and that

This is possible, since for a fixed R k , the preceding sum tends to zero as $$\varepsilon _{k} \rightarrow 0$$. Then for x ∈ E k we have

(4.2.25)

The inductive selection of the parameters can be schematically described as follows:

Observe that the construction of γ k gives for all s ≥ k + 1 the estimate

(4.2.26)

using (4.2.23) and the fact that the sequence $$\gamma _{k}$$ is decreasing.

We now prove (4.2.22). For $$x \in \mathbf{T}^{n}$$ write

from which it follows that

In view of (4.2.25), (4.2.24), and (4.2.26) for all $$x \in E_{k}$$, we obtain

which clearly implies (4.2.22). Setting

we deduce that $$\sup _{R>0}\big\vert B_{R}^{\frac{n-1} {2} }(f)(x)\big\vert \geq k$$ for all x in $$\bigcup _{r=k}^{\infty }E_{r}$$, and thus

for all x in

Since this set has full measure in $$\mathbf{T}^{n}$$, the required conclusion follows.

By taking $$\varepsilon _{1}$$ arbitrarily small (instead of picking $$\varepsilon _{1} = 1$$), we force f to be supported in an arbitrarily small neighborhood of the origin. $$\square $$

The previous argument shows that the Bochner–Riesz means $$B_{R}^{\alpha }$$ are badly behaved on $$L^{1}(\mathbf{T}^{n})$$ when $$\alpha = \tfrac{n-1} {2}$$. It follows that the “rougher” spherical Dirichlet means $$ D_{N }^{n }{\ast}f$$ (which correspond to $$\alpha = 0$$) are also ill behaved on $$L^{1}(\mathbf{T}^{n})$$. See Exercise 4.2.2.

Exercises

4.2.1. Using Theorem 4.2.1 construct a function F on $$\mathbf{T}^{n}$$ such that

for almost all $$(x_{1},\ldots,x_{n}) \in \mathbf{T}^{n}$$.

4.2.2. For any $$0 \leq \alpha < \infty $$ and R > 0 consider the Bochner–Riesz kernel

Use Exercise 4.1.8 to obtain that if for some $$x_{0} \in \mathbf{T}^{n}$$ we have

then for all $$\beta >\alpha$$ we have

Conclude that whenever $$0 \leq \alpha \leq \tfrac{n-1} {2}$$, the Bochner–Riesz means of order $$\alpha$$ of the function f constructed in the proof of Theorem 4.2.5, in particular the circular (spherical) Dirichlet means of this function, diverge a.e.

4.2.3. (a) Show that for M, N positive integers we have

(b) Prove that for some constant c > 0 we have

as $$N \rightarrow \infty $$.

[Hint: Part (b): Show that for $$x \in [-\tfrac{1} {2}, \tfrac{1} {2}]$$ we have

and use the result of Exercise 3.1.5.]

4.2.4. Given the integrable functions

show that $$\|f_{1} {\ast} D_{N}\|_{L^{1}} \rightarrow \infty $$ and $$\|f_{2} {\ast} D_{N}\|_{L^{1}} \rightarrow \infty $$ as $$N \rightarrow \infty $$.

[Hint: Let $$M_{j} = 2^{2^{2^{j}} }$$ or $$M_{j} = 2^{2^{j} }$$ depending on the situation. For fixed N let j N be the least integer j such that M j  > N. Then for j ≥ j N + 1 we have $$M_{j} \geq M_{j_{N}}^{2} > N^{2} \geq 2N + 1$$, hence $$\frac{M_{j}-N} {M_{j}+1} \geq \frac{1} {2}$$. Split the summation indices into the sets $$j \geq j_{N}$$ and $$j < j_{N}$$. Conclude that $$\|f_{1} {\ast} D_{N}\|_{L^{1}}$$ and $$\|f_{2} {\ast} D_{N}\|_{L^{1}}$$ tend to infinity as $$N \rightarrow \infty $$ using Exercise 4.2.3.]

4.3 Multipliers, Transference, and Almost Everywhere Convergence

In Chapter 2 we saw that bounded operators from $$L^{p}(\mathbf{R}^{n})$$ to $$L^{q}(\mathbf{R}^{n})$$ that commute with translations are given by convolution with tempered distributions on $$\mathbf{R}^{n}$$. In particular, when p = q, these tempered distributions have bounded Fourier transforms, called Fourier multipliers. Convolution operators that commute with translations can also be defined on the torus. These lead to Fourier multipliers on the torus.

4.3.1 Multipliers on the Torus

In analogy with the nonperiodic case, we could identify convolution operators on $$\mathbf{T}^{n}$$ with appropriate distributions on the torus; see Exercise 4.3.2 for an introduction to this topic. However, it is simpler to avoid this point of view and consider the study of multipliers directly, bypassing the discussion of distributions on the torus.

For $$h \in \mathbf{T}^{n}$$ we define the translation operator $$\tau ^{h}$$ acting on a periodic function f as follows: $$\tau ^{h}(f)(x) = f(x - h)$$ for $$x \in \mathbf{T}^{n}$$. We say that a linear operator T acting on functions on the torus commutes with translations if for all $$h \in \mathbf{T}^{n}$$ we have $$\tau ^{h}(T(f))(x) = T(\tau ^{h}f)(x)$$ for almost all $$x \in \mathbf{T}^{n}$$.

Theorem 4.3.1.

Suppose that T is a linear operator that commutes with translations and maps $$L^{p}(\mathbf{T}^{n})$$ to $$L^{q}(\mathbf{T}^{n})$$ for some $$1 \leq p,q \leq \infty $$ . Then there exists a bounded sequence $$\{a_{m}\}_{m\in \mathbf{Z}^{n}}$$ such that

(4.3.1)

for all $$f \in \mathcal{C}^{\infty }(\mathbf{T}^{n})$$ . Moreover, we have

Proof.

Consider the functions $$e_{m}(x) = e^{2\pi im\cdot x}$$ defined on $$\mathbf{T}^{n}$$ for m in $$\mathbf{Z}^{n}$$. Since $$T$$ commutes with translations, for every $$h \in \mathbf{T}^{n}$$ there is a subset F h of $$\mathbf{T}^{n}$$ of full measure such that

for every x ∈ F h . Note that

Therefore there exists an $$x_{0} \in \mathbf{T}^{n}$$ such that $$\vert \{h \in \mathbf{T}^{n}:\,\, x_{0} \in F_{h}\}\vert = 1$$. It follows that for almost all $$h \in \mathbf{T}^{n}$$ we have $$T(e_{m})(x_{0} - h) = e^{-2\pi im\cdot h}T(e_{m})(x_{0})$$. Replacing x ₀ − h by x, we obtain

(4.3.2)

for almost all $$x \in \mathbf{T}^{n}$$, where we set $$a_{m} = e^{-2\pi im\cdot x_{0}}T(e_{m})(x_{0})$$, for $$m \in \mathbf{Z}^{n}$$. Taking $$L^{q}$$ norms in (4.3.2), we deduce $$\vert a_{m}\vert =\| T(e_{m})\|_{L^{q}} \leq \| T\|_{L^{p}\rightarrow L^{q}}$$, and thus a m is bounded. Moreover, since $$T(e_{m}) = a_{m}e_{m}$$ for all m in $$\mathbf{Z}^{n}$$, it follows that (4.3.1) holds for all trigonometric polynomials. By density this extends to all $$f \in \mathcal{C}^{\infty }(\mathbf{T}^{n})$$ and the theorem is proved. □ 

Definition 4.3.2.

Let $$1 \leq p,q \leq \infty $$. We call a bounded sequence $$\{a_{m}\}_{m\in \mathbf{Z}^{n}}$$ an $$(L^{p},L^{q})$$ multiplier if the corresponding operator given by (4.3.1) maps $$L^{p}(\mathbf{T}^{n})$$ to $$L^{q}(\mathbf{T}^{n})$$. If p = q, $$(L^{p},L^{p})$$ multipliers are simply called L p multipliers. When $$1 \leq p < \infty $$, the space of all L p multipliers on $$\mathbf{T}^{n}$$ is denoted by $$\mathcal{M}_{p}(\mathbf{Z}^{n})$$. This notation follows the convention that $$\mathcal{M}_{p}(\widehat{G})$$ denote the space of $$L^{p}$$ multipliers on $$L^{p}(G)$$, where G is a locally compact group and $$\widehat{G}$$ is its dual group. The norm of an element {a m } in $$\mathcal{M}_{p}(\mathbf{Z}^{n})$$ is the norm of the operator T given by (4.3.1) from $$L^{p}(\mathbf{T}^{n})$$ to itself. This norm is denoted by $$\big\|\{a_{m}\}\big\|_{\mathcal{M}_{p}}$$.

We now examine some special cases. We begin with the case p = q = 2. As expected, it turns out that $$\mathcal{M}_{2}(\mathbf{Z}^{n}) =\ell ^{\infty }(\mathbf{Z}^{n})$$.

Theorem 4.3.3.

A linear operator T that commutes with translations maps $$L^{2}(\mathbf{T}^{n})$$ to itself if and only if there exists a sequence $$\{a_{m}\}_{m\in \mathbf{Z}^{n}}$$ in $$\ell^{\infty }(\mathbf{Z}^{n})$$ such that

(4.3.3)

for all $$f \in \mathcal{C}^{\infty }(\mathbf{T}^{n})$$ . Moreover, in this case we have $$\big\|T\big\|_{L^{2}\rightarrow L^{2}} =\big\|\{ a_{m}\}_{m}\big\|_{\ell^{\infty }}$$ .

Proof.

The existence of such a sequence is guaranteed by Theorem 4.3.1, which also gives $$\|\{a_{m}\}_{m}\|_{\ell^{\infty }} \leq \| T\|_{L^{2}\rightarrow L^{2}}$$. Conversely, any operator of the form (4.3.3) satisfies

and thus $$\|T\|_{L^{2}\rightarrow L^{2}} \leq \|\{ a_{m}\}_{m}\|_{\ell^{\infty }}$$. □ 

We continue with the case p = q = 1. Recall the definition of a finite Borel measure on $$\mathbf{T}^{n}$$. Given such a measure μ, its Fourier coefficients are defined by

Clearly, all the Fourier coefficients of the measure μ are bounded by the total variation $$\big\|\mu \big\|$$ of μ. See Exercise 4.3.3 for basic properties of Fourier transforms of distributions on the torus.

Theorem 4.3.4.

A linear operator T that commutes with translations maps $$L^{1}(\mathbf{T}^{n})$$ to itself if and only if there exists a finite Borel measure μ on the torus such that

(4.3.4)

for all $$f \in \mathcal{C}^{\infty }(\mathbf{T}^{n})$$ . Moreover, in this case we have $$\|T\|_{L^{1}\rightarrow L^{1}} =\|\mu \|$$ . In other words, $$\mathcal{M}_{1}(\mathbf{Z}^{n})$$ is the set of all sequences given by Fourier coefficients of finite Borel measures on $$\mathbf{T}^{n}$$ .

Proof.

Fix $$f \in L^{1}(\mathbf{T}^{n})$$. If (4.3.4) is valid, then $$\widehat{T(f)}(m) =\widehat{ f}(m)\widehat{\mu }(m)$$ for all $$m \in \mathbf{Z}^{n}$$. But Exercise 4.3.3 gives that $$\widehat{f {\ast}\mu } (m) =\widehat{ f}(m)\widehat{\mu }(m)$$ for all $$m \in \mathbf{Z}^{n}$$; therefore, the integrable functions f ∗μ and T(f) have the same Fourier coefficients, and they must be equal. Thus T(f) = f ∗μ, which implies that T is bounded on $$L^{1}$$ and $$\|T(f)\|_{L^{1}} \leq \|\mu \|\| f\|_{L^{1}}$$.

To prove the converse direction, we suppose that T commutes with translations and maps $$L^{1}(\mathbf{T}^{n})$$ to itself. We recall the Poisson kernel $$P_{\varepsilon }$$ defined on $$\mathbf{T}^{n}$$, which can be expressed in the following two ways:

(4.3.5)

for all $$x \in \mathbf{T}^{n}$$, in view of the identity obtained in (3.​2.​4). The preceding identity says that $$P_{\varepsilon } \geq 0$$; hence, $$\|P_{\varepsilon }\|_{L^{1}} =\int _{\mathbf{T}^{n}}P_{\varepsilon }(x)\,dx$$. Integrating the first series in (4.3.5) over $$\mathbf{T}^{n}$$ we conclude that $$\|P_{\varepsilon }\|_{L^{1}(\mathbf{T}^{n})} = 1$$. The boundedness of $$T$$ now gives

for all $$\varepsilon > 0$$. The Banach–Alaoglu theorem implies that there exist a sequence $$\varepsilon _{j} \downarrow 0$$ and a finite Borel measure $$\mu$$ on $$\mathbf{T}^{n}$$ such that $$T(P_{\varepsilon _{j}})$$ tends to μ weakly as $$j \rightarrow \infty $$. This means that for all continuous functions g on $$\mathbf{T}^{n}$$ we have

(4.3.6)

It follows from (4.3.6) that for all g continuous on $$\mathbf{T}^{n}$$ we have

Since, by the Riesz representation theorem we have that the norm of the linear functional

on the space of continuous functions $$\mathcal{C}(\mathbf{T}^{n})$$ is ∥ μ ∥ , it follows that

(4.3.7)

It remains to prove that T has the form given in (4.3.4). By Theorem 4.3.1 we have that there exists a bounded sequence $$\{a_{m}\}$$ on $$\mathbf{Z}^{n}$$ such that (4.3.1) is satisfied. Taking $$g(x) = e^{-2\pi ik\cdot x}$$ in (4.3.6) and using the representation for T in (4.3.1), we obtain

This proves assertion (4.3.4). It follows from (4.3.4) that T(f) = f ∗μ and thus $$\|T\|_{L^{1}\rightarrow L^{1}} \leq \|\mu \|$$. This fact combined with (4.3.7) gives $$\|T\|_{L^{1}\rightarrow L^{1}} =\|\mu \|$$. □ 

Remark 4.3.5.

It is not hard to see that most basic properties of the space $$\mathcal{M}_{p}(\mathbf{R}^{n})$$ of $$L^{p}$$ Fourier multipliers on $$\mathbf{R}^{n}$$ are also valid for $$\mathcal{M}_{p}(\mathbf{Z}^{n})$$. In particular, $$\mathcal{M}_{p}(\mathbf{Z}^{n})$$ is a closed subspace of $$\ell^{\infty }(\mathbf{Z}^{n})$$ and thus a Banach space itself. Moreover, sums, scalar multiples, and products of elements of $$\mathcal{M}_{p}(\mathbf{Z}^{n})$$ are also in $$\mathcal{M}_{p}(\mathbf{Z}^{n})$$, which makes this space a Banach algebra. As in the nonperiodic case, we also have $$\mathcal{M}_{p}(\mathbf{Z}^{n}) = \mathcal{M}_{p'}(\mathbf{Z}^{n})$$ when $$1 < p < \infty $$.

4.3.2 Transference of Multipliers

It is clear by now that multipliers on $$L^{1}(\mathbf{T}^{n})$$ and $$L^{1}(\mathbf{R}^{n})$$ are very similar, and the same is true for $$L^{2}(\mathbf{T}^{n})$$ and $$L^{2}(\mathbf{R}^{n})$$. These similarities became obvious when we characterized L ¹ and L ² multipliers on both $$\mathbf{R}^{n}$$ and $$\mathbf{T}^{n}$$. So far, it is not known if a nontrivial characterization of $$\mathcal{M}_{p}(\mathbf{R}^{n})$$ exists, but we might ask whether this space is related to $$\mathcal{M}_{p}(\mathbf{Z}^{n})$$. There are several connections of this type and there are general ways to produce multipliers on the torus from multipliers on $$\mathbf{R}^{n}$$ and vice versa. General methods of this sort are called transference of multipliers .

We begin with a useful definition.

Definition 4.3.6.

Let $$t_{0} \in \mathbf{R}^{n}$$. A bounded function b on $$\mathbf{R}^{n}$$ is called regulated at the point t ₀ if

(4.3.8)

The function b is called regulated if it is regulated at every $$t_{0} \in \mathbf{R}^{n}$$.

Clearly, if t ₀ is a Lebesgue point of b, then b is regulated at t ₀. In particular, this is the case if b is continuous at t ₀. If b(t ₀) = 0, condition (4.3.8) also holds when $$b(t_{0} - t) = -b(t_{0} + t)$$ whenever $$\vert t\vert \leq \varepsilon$$ for some $$\varepsilon > 0$$; for instance the function $$b(t) = -i\,\mbox{ sgn}\,(t - t_{0})$$ has this property.

An example of a regulated function is the following modification of the characteristic function of the cube [−1, 1] n

with the understanding that points are zero-dimensional.

The first transference result we discuss is the following.

Theorem 4.3.7.

Suppose that b is a regulated function at every point $$m \in \mathbf{Z}^{n}$$ and that b lies in $$\mathcal{M}_{p}(\mathbf{R}^{n})$$ for some $$1 < p < \infty $$ . Then the sequence $$\{b(m)\}_{m\in \mathbf{Z}^{n}}$$ is in $$\mathcal{M}_{p}(\mathbf{Z}^{n})$$ and moreover,

If b is regulated everywhere, then for all R > 0 the sequences $$\{b(m/R)\}_{m\in \mathbf{Z}^{n}}$$ are in $$\mathcal{M}_{p}(\mathbf{Z}^{n})$$ and we have

The second conclusion of the theorem is a consequence of the first, since for a given R > 0 the function $$b(\xi /R)$$ is regulated on $$\mathbf{Z}^{n}$$ and has the same $$\mathcal{M}_{p}(\mathbf{R}^{n})$$ norm as $$b(\xi )$$. Before we prove this result, we state and prove a couple of lemmas.

Lemma 4.3.8.

Suppose that the function b on $$\mathbf{R}^{n}$$ is regulated at the point $$x_{0}$$ . Let $$K_{\varepsilon }(x) =\varepsilon ^{-n}e^{-\pi \vert x/\varepsilon \vert ^{2} }$$ for $$\varepsilon > 0$$ . Then we have that $$(b {\ast} K_{\varepsilon })(x_{0}) \rightarrow b(x_{0})$$ as $$\varepsilon \rightarrow 0$$ .

Proof.

For r > 0 define the function

Let η > 0. Since b is regulated at x ₀ there is a δ > 0 such that for r ≤ δ we have $$\vert F_{x_{0}}(r)\vert \leq \eta$$. Fix such a $$\delta$$ and write

where

and

For our given $$\eta > 0$$ there is an $$\varepsilon _{0} > 0$$ such that for $$\varepsilon <\varepsilon _{0}$$ we have

Via an integration by parts $$A_{2}^{\varepsilon }$$ can be written as

where we set $$C_{n} =\sup _{t>0}t^{n}e^{-\pi t^{2} }$$ and $$C_{n}' = 2\pi \int _{0}^{\infty }r^{n+1}e^{-\pi r^{2} }dr$$. Then for $$\varepsilon <\varepsilon _{0}$$ we have $$\vert (b {\ast} K_{\varepsilon })(x_{0} - b(x_{0})\vert < (C_{n} + C_{n}' + 1)\eta$$, thus $$(b {\ast} K_{\varepsilon })(x_{0}) \rightarrow b(x_{0})$$ as $$\varepsilon \rightarrow 0$$. □ 

Lemma 4.3.9.

Let T be the operator on $$\mathbf{R}^{n}$$ whose multiplier is b(ξ), and let S be the operator on $$\mathbf{T}^{n}$$ whose multiplier is the sequence $$\{b(m)\}_{m\in \mathbf{Z}^{n}}$$ . Assume that $$b(\xi )$$ is regulated at every point $$\xi = m \in \mathbf{Z}^{n}$$ . Suppose that P and Q are trigonometric polynomials on $$\mathbf{T}^{n}$$ and let $$L_{\varepsilon }(x) = e^{-\pi \varepsilon \vert x\vert ^{2} }$$ for $$x \in \mathbf{R}^{n}$$ and $$\varepsilon > 0$$ . Then the following identity is valid whenever $$\alpha,\beta > 0$$ and $$\alpha +\beta = 1$$:

(4.3.9)

Proof.

It suffices to prove the required assertion for $$P(x) = e^{2\pi im\cdot x}$$ and $$Q(x) = e^{2\pi ik\cdot x}$$, $$k,m \in \mathbf{Z}^{n}$$, since the general case follows from this case by linearity. In view of Parseval’s relation (Proposition 3.​2.​7 (3)), we have

(4.3.10)

On the other hand, using the identity in Theorem 2.​2.​14 (3), we obtain

(4.3.11)

Now if m = k, since $$\alpha +\beta = 1$$, the expression in (4.3.11) is equal to

(4.3.12)

which tends to b(m) in view of Lemma 4.3.8, since b is regulated at every point $$m \in \mathbf{Z}^{n}$$.

We now consider the case m ≠ k in (4.3.11). Since | m − k | ≥ 1, then every ξ in $$\mathbf{R}^{n}$$ must satisfy either | ξ − m | ≥ 1∕2 or | ξ − k | ≥ 1∕2. Therefore, the expression in (4.3.11) is controlled by

which is in turn controlled by

which tends to zero as $$\varepsilon \rightarrow 0$$. This proves that the expression in (4.3.10) is equal to the limit of the expression in (4.3.11) as $$\varepsilon \rightarrow 0$$. This completes the proof of Lemma 4.3.9 □ 

Proof (Theorem 4.3.7).

We are assuming that T maps $$L^{p}(\mathbf{R}^{n})$$ to itself and we need to show that S maps $$L^{p}(\mathbf{T}^{n})$$ to itself. We prove this using duality. For P and Q trigonometric polynomials, using Lemma 4.3.9, we have

provided for all continuous 1-periodic functions g on $$\mathbf{R}^{n}$$ we have that

(4.3.13)

Assuming (4.3.13) for the moment, we take the supremum over all trigonometric polynomials Q on $$\mathbf{T}^{n}$$ with L p′ norm at most 1 to obtain that S maps $$L^{p}(\mathbf{T}^{n})$$ to itself with norm at most $$\|T\|_{L^{p}\rightarrow L^{p}}$$, yielding the required conclusion.

We now prove (4.3.13). Use the Poisson summation formula to write the left-hand side of (4.3.13) as

where

as $$\varepsilon \rightarrow 0$$. This completes the proof of Theorem 4.3.7. □ 

We now obtain a converse of Theorem 4.3.7. If b(ξ) is a bounded function on $$\mathbf{R}^{n}$$ and the sequence $$\{b(m)\}_{m\in \mathbf{Z}^{n}}$$ is in $$\mathcal{M}_{p}(\mathbf{Z}^{n})$$, then we cannot necessarily obtain that b is in $$\mathcal{M}_{p}(\mathbf{R}^{n})$$, since such a conclusion would depend on the values of b on the integer lattice, which is a set of measure zero. However, a converse can be formulated if we assume that for all R > 0, the sequences $$\{b(m/R)\}_{m\in \mathbf{Z}^{n}}$$ are in $$\mathcal{M}_{p}(\mathbf{Z}^{n})$$ uniformly in R. Then we obtain that $$b(\xi /R)$$ is in $$\mathcal{M}_{p}(\mathbf{R}^{n})$$ uniformly in R > 0, which is equivalent to saying that $$b \in \mathcal{M}_{p}(\mathbf{R}^{n})$$, since dilations of multipliers on $$\mathbf{R}^{n}$$ do not affect their norms (see Proposition 2.​5.​14). These remarks can be precisely expressed in the following theorem.

Theorem 4.3.10.

Suppose that $$b(\xi )$$ is a bounded function defined on $$\mathbf{R}^{n}$$ which is Riemann integrable over any cube. Suppose that the sequences $$\{b(\tfrac{m} {R} )\}_{m\in \mathbf{Z}^{n}}$$ are in $$\mathcal{M}_{p}(\mathbf{Z}^{n})$$ uniformly in $$R > 0$$ for some $$1 < p < \infty $$ . Then b is in $$\mathcal{M}_{p}(\mathbf{R}^{n})$$ and we have

(4.3.14)

Proof.

Suppose that f and g are smooth functions with compact support on $$\mathbf{R}^{n}$$. Then there is an R ₀ > 0 such that for R ≥ R ₀, the functions $$x\mapsto f(Rx)$$ and $$x\mapsto g(Rx)$$ are supported in [−1∕2, 1∕2] n . We define periodic functions

on $$\mathbf{T}^{n}$$. Observe that the mth Fourier coefficient of F R is $$\widehat{F_{R}}(m) = R^{-n}\widehat{f}(m/R)$$ and that of $$G_{R}$$ is $$\widehat{G_{R}}(m) = R^{-n}\widehat{g}(m/R)$$.

Now for $$R \geq R_{0}$$ we have

(4.3.15)

(4.3.16)

Since b is bounded and Riemann integrable over any cube in $$\mathbf{R}^{n}$$, the function $$b(\xi )\widehat{f}(\xi )\overline{\widehat{g}(\xi )}$$ is Riemann integrable over $$\mathbf{R}^{n}$$. The expressions in (4.3.15) are sums associated with the partition $$\{[\tfrac{m} {R}, \tfrac{m+1} {R} )^{n}\}_{ m\in \mathbf{Z}^{n}}$$ of $$\mathbf{R}^{n}$$ which tend to

as $$R \rightarrow \infty $$ by the definition of the Riemann integral. We deduce that the absolute value of

is bounded by the expression in (4.3.16). This proves the theorem via duality. □ 

4.3.3 Applications of Transference

Having established two main transference theorems, we turn to an application.

Corollary 4.3.11.

Let $$1 < p < \infty $$, $$f \in L^{p}(\mathbf{T}^{n})$$ , and $$\alpha \geq 0$$ . Then

1.

$$\big\|D_{R}^{n} {\ast} f - f\big\|_{L^{p}(\mathbf{T}^{n})} \rightarrow 0$$ as $$R \rightarrow \infty $$ if and only if $$\chi _{[-1,1]^{n}} \in \mathcal{M}_{p}(\mathbf{R}^{n})$$ .

2.

$$\big\| D_{R }^{n } {\ast}f - f\big\|_{L^{p}(\mathbf{T}^{n})} \rightarrow 0$$ as $$R \rightarrow \infty $$ if and only if $$\chi _{B(0,1)} \in \mathcal{M}_{p}(\mathbf{R}^{n})$$ .

3.

$$\big\|B_{R}^{\alpha }(f) - f\big\|_{L^{p}(\mathbf{T}^{n})} \rightarrow 0$$ as $$R \rightarrow \infty $$ if and only if $$(1 -\vert \xi \vert ^{2})_{+}^{\alpha } \in \mathcal{M}_{p}(\mathbf{R}^{n})$$ .

Proof.

First observe that in view of Corollary 4.1.3, the assertions on the left in (a), (b), and (c) are equivalent to the statements

for some constant $$0 < C_{p} < \infty $$ and all f in $$L^{p}(\mathbf{T}^{n})$$. These statements can be rephrased as

If these statements hold, then Theorem 4.3.10 gives that the functions $$\chi _{[-1,1]^{n}}(\xi )$$, $$\chi _{B(0,1)}(\xi )$$, and $$(1 -\vert \xi \vert ^{2})_{+}^{\alpha }$$ lie in $$\mathcal{M}_{p}(\mathbf{R}^{n})$$.

To prove the converse implication, for any given $$R' \in \mathbf{R}^{+}\setminus \{\vert m\vert:\,\, m \in \mathbf{Z}^{n}\}$$, the functions $$\chi _{[-1,1]^{n}}(\xi /R')$$, $$\chi _{B(0,1)}(\xi /R')$$ are Riemann integrable over $$\mathbf{R}^{n}$$ and are regulated (actually continuous) at every point in $$\mathbf{Z}^{n}$$. Moreover, the function $$(1 -\vert \xi \vert ^{2})_{+}^{\alpha }$$ is continuous, regulated, and Riemann integrable over $$\mathbf{R}^{n}$$. Then the hypotheses of Theorem 4.3.7 are satisfied and its conclusion yields that

(4.3.17)

(4.3.18)

Notice that the first and second estimates are uniform in R′, so one may insert a supremum over $$R' \in \mathbf{R}^{+}\setminus \{\vert m\vert:\,\, m \in \mathbf{Z}^{n}\}$$ in (4.3.17) and (4.3.18). To replace R′ by a general $$R \in \mathbf{Z}^{+}$$ simply notice that for any R > 0 there is an $$R' \in \mathbf{R}^{+}\setminus \{\vert m\vert:\,\, m \in \mathbf{Z}^{n}\}$$ such that

for any $$f \in L^{p}(\mathbf{T}^{n})$$. Then using (4.3.17) we obtain

and likewise for $$ D_{R }^{n }$$. □ 

4.3.4 Transference of Maximal Multipliers

We now prove a theorem concerning maximal multipliers analogous to Theorems 4.3.7 and 4.3.10. This enables us to reduce problems related to almost everywhere convergence of Fourier series on the torus to problems of boundedness of maximal operators on $$\mathbf{R}^{n}$$.

Let $$b$$ be a bounded function defined on all of $$\mathbf{R}^{n}$$. For R > 0, we introduce the multiplier operators

(4.3.19)

(4.3.20)

initially defined for smooth functions with compact support f on $$\mathbf{R}^{n}$$ and smooth functions $$F$$ on $$\mathbf{T}^{n}$$.

We introduce the maximal operators

(4.3.21)

(4.3.22)

defined for smooth functions F on $$\mathbf{T}^{n}$$ and smooth functions with compact support f on $$\mathbf{R}^{n}$$. Let $$\tau ^{y}(b)(\xi ) = b(\xi -y)$$ be a translation operator defined for $$y \in \mathbf{R}^{n}$$. We have the following result concerning these operators.

Theorem 4.3.12.

Let b be a function defined on $$\mathbf{R}^{n}$$ . Suppose that b is bounded, regulated, Riemann integrable over any cube, and assume that for all $$\xi \in \mathbf{R}^{n}$$ the function $$t\mapsto b(\xi /t)$$ has only countably many discontinuities on $$\mathbf{R}^{+}$$ . Let $$1 < p < \infty $$ and $$C_{p} < \infty $$ , and suppose that b lies in $$\mathcal{M}_{p}(\mathbf{R}^{n})$$ . Let M b and N b be as in (4.3.21) and (4.3.22) . Then the following assertions are equivalent:

(4.3.23)

(4.3.24)

Proof.

Let $$\mathcal{F} =\{ t_{1},\ldots,t_{k}\}$$ be a finite subset of $$\mathbf{R}^{+}$$. We prove the claimed equivalences for the maximal operators

with constants that are uniform in the finite set $$\mathcal{F}$$. Then $$M_{b}^{\mathcal{F}}$$ may be viewed as an operator defined on the dense subspace $$\mathcal{C}^{\infty }(\mathbf{T}^{n})$$ of $$L^{p}(\mathbf{T}^{n})$$ and taking values in $$L^{p}(\mathbf{T}^{n},\ell^{\infty }(\mathcal{F}))$$, which is the dual space of $$L^{p'}(\mathbf{T}^{n},\ell^{1}(\mathcal{F}))$$. Likewise, $$N_{b}^{\mathcal{F}}$$ is defined on the dense subspace $$\mathcal{C}_{0}^{\infty }(\mathbf{R}^{n})$$ of $$L^{p}(\mathbf{R}^{n})$$ and takes values in $$L^{p}(\mathbf{R}^{n},\ell^{\infty }(\mathcal{F}))$$, which is the dual space of $$L^{p'}(\mathbf{R}^{n},\ell^{1}(\mathcal{F}))$$. Using duality, with respect to the complex inner product, estimates (4.3.23) and (4.3.24) are equivalent to the pair of inequalities

(4.3.25)

(4.3.26)

where $$g,f_{j} \in \mathcal{C}_{0}^{\infty }(\mathbf{R}^{n})$$, and $$G,F_{j} \in \mathcal{C}^{\infty }(\mathbf{T}^{n})$$. In proving the equivalence of (4.3.25) and (4.3.26), by density, we work with smooth functions with compact support g, f j and trigonometric polynomials G, F j .

Suppose that (4.3.25) holds and let $$f_{1},\ldots,f_{k},g$$ be smooth functions with compact support on $$\mathbf{R}^{n}$$. Then there is an R ₀ > 0 such that for R ≥ R ₀ the functions $$F_{j,R}(x) = f_{j}(Rx)$$ and $$G_{R}(x) = g(Rx)$$ are supported in $$[-1/2,1/2]^{n}$$ and thus they can be viewed as functions on $$\mathbf{T}^{n}$$ once they are periodized. Also, the $$m$$ th Fourier coefficient of $$F_{j,R}$$ is $$\widehat{F_{j,R}}(m) = R^{-n}\widehat{f_{j}}(m/R)$$ and that of $$G_{R}$$ is $$\widehat{G_{R}}(m) = R^{-n}\widehat{g}(m/R)$$. Since $$b$$ lies in $$\mathcal{M}_{p}(\mathbf{R}^{n})$$ we have

in view of Theorems 4.3.7 and 4.3.10, which are both applicable in view of the hypotheses of b.

As in the proof of Theorem 4.3.10, for $$R \geq R_{0}$$ we have

(4.3.27)

where we applied (4.3.25) in the first inequality above for the function $$\xi \mapsto b(\xi /R)$$, which has the same $$\mathcal{M}_{p}$$ norm as b.

Since b is bounded and Riemann integrable over any cube in $$\mathbf{R}^{n}$$, the functions $$b(\xi /t_{j})\widehat{f_{j}}(\xi )\overline{\widehat{g_{j}}(\xi )}$$ are Riemann integrable over $$\mathbf{R}^{n}$$. Realizing the limit of the partial sums in (4.3.27) when $$R \rightarrow \infty $$ as a Riemann integral, we obtain

and thus we showed that (4.3.25) implies (4.3.26).

We now turn to the converse. Assume that (4.3.26) holds. We will prove (4.3.25) for trigonometric polynomials and then by density we extend it to all $$\mathcal{C}^{\infty }$$ functions on $$\mathbf{T}^{n}$$. Expressing $$\widehat{g}$$ in terms of $$g$$ in (4.3.26) and taking the supremum in (4.3.26) over all $$\mathcal{C}_{0}^{\infty }$$ functions g with $$L^{p}$$ norm 1 we deduce that

(4.3.28)

Let $$P_{1},\ldots,P_{k}$$ and Q be trigonometric polynomials on $$\mathbf{T}^{n}$$. Set $$L_{\varepsilon }(x) = e^{-\pi \varepsilon \vert x\vert ^{2} }$$. Since $$b$$ is regulated at every point in $$\mathbf{R}^{n}$$, Lemma 4.3.9 gives

where we used Hölder’s inequality and (4.3.28) in the only inequality above and (4.3.13) in the last equality. Thus we obtain that (4.3.26) implies (4.3.25), and this completes the equivalence of boundedness of $$M_{b}^{\mathcal{F}}$$ and $$N_{b}^{\mathcal{F}}$$.

We now prove the claimed equivalence for the operators M b and N b . We first show that if $$M_{b}^{\mathcal{F}}$$ is bounded on $$(\mathcal{C}^{\infty }(\mathbf{T}^{n}),\|\cdot \|_{L^{p}})$$ with bound independent of the finite set $$\mathcal{F}$$, then $$M_{b}$$ is bounded on $$(\mathcal{C}^{\infty }(\mathbf{T}^{n}),\|\cdot \|_{L^{p}})$$.

For each $$\xi \in \mathbf{R}^{n}$$, let A ξ be the null subset of $$\mathbf{R}^{+}$$ such that $$t\mapsto b(\xi /t)$$ is continuous on $$\mathbf{R}^{+}\setminus A_{\xi }$$. We fix a function F in $$\mathcal{C}^{\infty }(\mathbf{T}^{n})$$, and we note that for each $$x \in \mathbf{T}^{n}$$ the function

(4.3.29)

is continuous on the set $$\mathbf{R}^{+}\setminus \bigcup _{m\in \mathbf{Z}^{n}}A_{m}$$. We pick a countable dense subset $$D'$$ of $$\mathbf{R}^{+}\setminus \bigcup _{m\in \mathbf{Z}^{n}}A_{m}$$, and we let $$D = D' \cup \bigcup _{m\in \mathbf{Z}^{n}}A_{m}$$. Then $$D$$ is a countable set and the Lebesgue monotone convergence theorem gives that

(4.3.30)

where $$\mathcal{F}_{k}$$ is an increasing sequence of finite sets whose union is D. Using that the function in (4.3.29) is continuous on $$\mathbf{R}^{+}\setminus D$$, we conclude that the supremum over t ∈ D in (4.3.30) can be replaced by the supremum over $$t \in \mathbf{Z}^{+}$$ (Exercise 4.3.7).

Assume now that $$N_{b}^{\mathcal{F}}$$ is bounded on $$(\mathcal{C}_{0}^{\infty }(\mathbf{R}^{n}),\|\cdot \|_{L^{p}})$$ with bound independent of the finite set $$\mathcal{F}$$. We show that N b is bounded on $$(\mathcal{C}_{0}^{\infty }(\mathbf{R}^{n}),\|\cdot \|_{L^{p}})$$. Let f be in $$\mathcal{C}_{0}^{\infty }(\mathbf{R}^{n})$$. We have that the map

(4.3.31)

is a continuous function on $$\mathbf{R}^{+}$$ since $$\widehat{f}$$ is continuous. Thus the estimate

(4.3.32)

for a countable dense subset D of $$\mathbf{R}^{+}$$ (such as D = Q ⁺) can be easily extended by replacing the supremum over D by the supremum over $$\mathbf{R}^{+}$$. And estimate (4.3.32) for D = Q ⁺ follows from the corresponding estimate on finite sets via the Lebesgue monotone convergence theorem. □ 

Remark 4.3.13.

Under the hypotheses of Theorem 4.3.12, the following two inequalities are also equivalent:

(4.3.33)

(4.3.34)

with $$C_{p}' \leq C_{p} \leq C(n,p)C_{p}'$$ for some other constant C(n, p). Indeed, Exercise 1.4.12 gives that the pair of inequalities (4.3.33) and (4.3.34) is equivalent to the pair of inequalities

(4.3.35)

(4.3.36)

where L p′, 1 is the Lorentz space and $$f_{j} \in \mathcal{C}_{0}^{\infty }(\mathbf{R}^{n})$$ and $$F_{j} \in \mathcal{C}^{\infty }(\mathbf{T}^{n})$$.

Now (4.3.36) follows from (4.3.35) just like (4.3.26) follows from (4.3.25) with the only exception being that Hölder’s inequality for L p and L p′ is replaced by Hölder’s inequality for $$L^{p,\infty }$$ and L p′, 1 and we use that $$\|g\|_{L^{p,\infty }} \leq \| g\|_{L^{p}}$$. Conversely, assuming (4.3.36), in order to prove (4.3.35) it will suffice to know that

(4.3.37)

For this we refer to Exercise 4.3.6.

4.3.5 Applications to Almost Everywhere Convergence

As an application of the preceding results, we relate the almost everywhere convergence of Fourier series of functions on $$\mathbf{T}^{1}$$ with the almost everywhere convergence of Fourier integrals of functions on $$\mathbf{R}$$. In this subsection we show that the following two results are equivalent:

Theorem 4.3.14.

For every $$1 < p < \infty $$ there exists a finite constant C p such that for all $$F \in \mathcal{C}^{\infty }(\mathbf{T}^{1})$$ we have

(4.3.38)

Theorem 4.3.15.

For every $$1 < p < \infty $$ there exists a finite constant C p such that for all $$f \in \mathcal{C}_{0}^{\infty }(\mathbf{R})$$ we have

(4.3.39)

where

is the Carleson operator .

As a consequence of Theorem 4.3.14, we obtain that for any $$F \in L^{p}(\mathbf{T}^{1})$$, we have

for almost every x ∈ [0, 1].

Theorem 4.3.14 can be proved directly, but we do not pursue this here. Instead, we show the equivalence of the two theorems and refer the interested reader to [131], which contains the proof of Theorem 4.3.15.

We observe that both operators $$F\mapsto \mathcal{C}_{{\ast}}(F) =\sup _{N>0}\vert F {\ast} D_{N}\vert $$ and $$f\mapsto \mathcal{C}_{{\ast}{\ast}}(f)$$ are sublinear and take nonnegative values. Thus they satisfy the inequalities

for all $$F,G$$ in $$\mathcal{C}^{\infty }(\mathbf{T}^{1})$$ and $$f,g$$ in $$\mathcal{C}_{0}^{\infty }(\mathbf{R})$$. Then, by density (see the argument in the proof of Theorem 1.​4.​19 or Exercise 1.4.17), they admit bounded extensions to $$L^{p}(\mathbf{T}^{1})$$ and $$L^{p}(\mathbf{R})$$, respectively, so that (4.3.38) and (4.3.39) hold for all $$F \in L^{p}(\mathbf{T}^{1})$$ and $$f \in L^{p}(\mathbf{R})$$.

Next, we discuss the details of the transference argument that claims the equivalence of Theorems 4.3.14 and 4.3.15.

Consider the following function defined on $$\mathbf{R}$$:

(4.3.40)

Then b is bounded and Riemann integrable over any interval, and is easily seen to be regulated; also, given any $$x \in \mathbf{R}$$, the function $$t\mapsto b(x/t)$$ is discontinuous only for $$t \in \{ x,-x\}$$.

Let S b, R be as in (4.3.19), where b is defined in (4.3.40). We note that inequality (4.3.38) is equivalent to

(4.3.41)

for all $$F \in \mathcal{C}^{\infty }(\mathbf{T}^{1})$$, where $$\{D_{R}\}_{R>0}$$ is the family of Dirichlet kernels as defined in (3.​1.​16), depending on the continuous parameter R. Indeed, we have

(4.3.42)

Since $$\sup _{R>0}\vert \widehat{F}(\pm R)\vert \leq \| F\|_{L^{1}} \leq \| F\|_{L^{p}}$$, it follows that if (4.3.38) holds, then (4.3.41) also holds with $$C_{p}' = C_{p} + 1$$.

The only hypothesis of Theorem 4.3.12 missing is that b lies in $$\mathcal{M}_{p}(\mathbf{R})$$. We obtain this from the fact that $$\sup _{R>0}\|b(\cdot /R)\|_{\mathcal{M}_{p}(\mathbf{Z})} < \infty $$ via Theorem 4.3.10, since

(4.3.43)

where the last estimate follows from Proposition 4.1.6, Theorem 4.1.7, and Corollary 4.1.3. The preceding equality is due to the fact that $$D_{R} = D_{R+\varepsilon }$$ whenever $$0 <\varepsilon < 1$$.

Now all hypotheses of Theorem 4.3.12 are valid. As a consequence we obtain the equivalence of the boundedness of the the maximal operator

on $$L^{p}(\mathbf{R})$$ and of

on $$L^{p}(\mathbf{T}^{1})$$. But in view of (4.3.42) and of the fact that $$\sup _{R>0}\vert \widehat{F}(\pm R)\vert \leq \| F\|_{L^{p}}$$, the $$L^{p}$$ boundedness of M b is equivalent to the $$L^{p}$$ boundedness of $$\mathcal{C}_{{\ast}}$$ on $$L^{p}(\mathbf{T}^{1})$$. This discussion concludes the equivalence of Theorems 4.3.14 and 4.3.15.

4.3.6 Almost Everywhere Convergence of Square Dirichlet Means

The extension of Theorem 4.3.14 to higher dimensions is a rather straightforward consequence of the one-dimensional result.

Theorem 4.3.16.

For every $$1 < p < \infty $$ , there exists a finite constant $$C_{p,n}$$ such that for all $$f \in L^{p}(\mathbf{T}^{n})$$ we have

(4.3.44)

and consequently

for almost every $$x \in \mathbf{T}^{n}$$ and $$f \in L^{p}(\mathbf{T}^{n})$$ .

Proof.

We prove Theorem 4.3.16 when n = 2. Fix a p with $$1 < p < \infty $$. Since the Riesz projection P ₊ is bounded on $$L^{p}(\mathbf{T}^{1})$$ (Theorem 4.1.7 and identity (4.1.9)), applying Theorem 4.3.10 with $$b(\xi ) =\chi _{(0,\infty )}$$, we obtain that that the function $$\chi _{(0,\infty )}$$ is in $$\mathcal{M}_{p}(\mathbf{R})$$. It follows that the characteristic function of the half-space ξ ₁ > 0 in R ² lies in $$\mathcal{M}_{p}(\mathbf{R}^{2})$$. Since rotations and translations of multipliers preserve their $$\mathcal{M}_{p}$$ norms (Proposition 2.​5.​14), it follows that the characteristic function of any half space created by a line in $$\mathbf{R}^{2}$$ lies in $$\mathcal{M}_{p}(\mathbf{R}^{2})$$ with a fixed norm. The product of three multipliers is a multiplier (Proposition 2.​5.​13); thus the characteristic function of the triangle T created by the lines $$\xi _{2} =\xi _{1} -\frac{1} {4}$$, $$\xi _{2} = -\xi _{1} -\frac{1} {4}$$, $$\xi _{2} = L + \frac{1} {4}$$ lies also in $$\mathcal{M}_{p}(\mathbf{R}^{2})$$ with norm independent of $$L \in \mathbf{Z}^{+}$$. The regulated function

is a.e. equal to the characteristic function of T. Thus Theorem 4.3.7 gives that the restriction of $$\sigma$$ on Z ², i.e., the sequence $$\{a_{m_{1},m_{2}}\}_{m_{1},m_{2}}$$ defined by $$a_{m_{1},m_{2}} = 1$$ when $$\vert m_{1}\vert \leq \vert m_{2}\vert \leq L$$ and zero otherwise, lies in $$\mathcal{M}_{p}(\mathbf{Z}^{2})$$ with norm independent of L in $$\mathbf{Z}^{+}$$. This means that for some constant $$B_{p}$$ we have the following inequality for all f in $$L^{p}(\mathbf{T}^{2})$$:

(4.3.45)

where B p is independent of $$L \in \mathbf{Z}^{+}$$. There is also a a version of (4.3.45), proved similarly, in which $$\vert m_{1}\vert \leq \vert m_{2}\vert $$ is replaced by the strict inequality $$\vert m_{1}\vert < \vert m_{2}\vert $$.

Now let $$1 < p < \infty $$, $$L \in \mathbf{Z}^{+}$$, and $$f \in L^{p}(\mathbf{T}^{2})$$. For fixed $$x_{1} \in \mathbf{T}^{1}$$ define

and for fixed $$x_{2} \in \mathbf{T}^{1}$$ define

We have

where we used Theorem 4.3.14 in the penultimate inequality and estimate (4.3.45) in the last inequality. Since the last estimate we obtained is independent of $$L \in \mathbf{Z}^{+}$$, letting $$L \rightarrow \infty $$ and applying Fatou’s lemma, we obtain the conclusion (4.3.44) for n = 2. When n ≥ 3 the idea of the proof is similar, but the notation a bit more cumbersome. □ 

Exercises

4.3.1. Let $$\alpha \geq 0$$. Prove that the function $$(1 -\vert \xi \vert ^{2})_{+}^{\alpha }$$ is in $$\mathcal{M}_{p}(\mathbf{R}^{n})$$ if and only if the function $$(1 -\vert \xi \vert )_{+}^{\alpha }$$ is in $$\mathcal{M}_{p}(\mathbf{R}^{n})$$.

[Hint: Use that smooth functions with compact support lie in $$\mathcal{M}_{p}(\mathbf{R}^{n})$$.]

4.3.2. The purpose of this exercise is to introduce distributions on the torus. The set of test functions on the torus is $$\mathcal{C}^{\infty }(\mathbf{T}^{n})$$ equipped with the following topology. Given f j , f in $$\mathcal{C}^{\infty }(\mathbf{T}^{n})$$, we say that $$f_{j} \rightarrow f$$ in $$\mathcal{C}^{\infty }(\mathbf{T}^{n})$$ if

Under this notion of convergence, $$\mathcal{C}^{\infty }(\mathbf{T}^{n})$$ is a topological vector space with topology induced by the family of seminorms $$\rho _{\alpha }(\varphi ) =\sup _{x\in \mathbf{T}^{n}}\vert (\partial ^{\alpha }f)(x)\vert $$, where $$\alpha$$ ranges over all multi-indices. The dual space of $$\mathcal{C}^{\infty }(\mathbf{T}^{n})$$ under this topology is the set of all distributions on $$\mathbf{T}^{n}$$ and is denoted by $$\mathcal{D}'(\mathbf{T}^{n})$$. The definition implies that for u j and u in $$\mathcal{D}'(\mathbf{T}^{n})$$ we have u j  → u in $$\mathcal{D}'(\mathbf{T}^{n})$$ if and only if

The following operations can be defined on elements of $$\mathcal{D}'(\mathbf{T}^{n})$$: differentiation (as in Definition 2.3.5), translation and reflection (as in Definition 2.3.10), convolution with a $$\mathcal{C}^{\infty }$$ function (as in Definition 2.3.12), multiplication by a $$\mathcal{C}^{\infty }$$ function (as in Definition 2.3.14), the support of a distribution (as in Definition 2.​3.​15). Use the same ideas as in $$\mathbf{R}^{n}$$ to prove the following:

(a) Prove that if $$u \in \mathcal{D}'(\mathbf{T}^{n})$$ and $$f \in \mathcal{C}^{\infty }(\mathbf{T}^{n})$$, then $$(f {\ast} u)(x) =\langle u,\tau ^{x}(\widetilde{f}\,)\rangle$$ is a $$\mathcal{C}^{\infty }$$ function.

(b) In contrast to $$\mathbf{R}^{n}$$, the convolution of two distributions on $$\mathbf{T}^{n}$$ can be defined. For $$u,v \in \mathcal{D}'(\mathbf{T}^{n})$$ and $$f \in \mathcal{C}^{\infty }(\mathbf{T}^{n})$$ define

Check that convolution of distributions on $$\mathcal{D}'(\mathbf{T}^{n})$$ is associative, commutative, and distributive.

(c) Prove the analogue of Proposition 2.​3.​22, i.e., that $$\mathcal{C}^{\infty }(\mathbf{T}^{n})$$ is dense in $$\mathcal{D}'(\mathbf{T}^{n})$$.

4.3.3. For $$u \in \mathcal{D}'(\mathbf{T}^{n})$$ and $$m \in \mathbf{Z}^{n}$$ define the Fourier coefficient $$\widehat{u}(m)$$ by

Prove properties (1), (2), (4), (5), (6), (8), (9), (11), and (12) of Proposition 2.3.21 regarding the Fourier coefficients of distributions on the circle. Moreover, prove that for any u, v in $$\mathcal{D}'(\mathbf{T}^{n})$$ we have $$(u {\ast} v)\,\,\widehat{\,}\,\,(m) =\widehat{ u}(m)\,\widehat{v}(m)$$. In particular, this is valid for finite Borel measures.

4.3.4. Let μ be a finite Borel measure on $$\mathbf{R}^{n}$$ and let ν be the periodization of μ, that is, ν is a measure on $$\mathbf{T}^{n}$$ defined by

for all measurable subsets A of $$\mathbf{T}^{n}$$. Prove that the restriction of the Fourier transform of μ on $$\mathbf{Z}^{n}$$ coincides with the sequence of the Fourier coefficients of the measure $$\nu$$.

4.3.5. Let v n be the volume of the unit ball in $$\mathbf{R}^{n}$$ and $$e_{1} = (1,0,\ldots,0)$$. Prove that

Conclude that the function

is regulated.

4.3.6. Let $$L_{\varepsilon }(x) = e^{-\pi \varepsilon \vert x\vert ^{2} }$$ be defined for $$\varepsilon > 0$$ and $$x \in \mathbf{R}^{n}$$ and let $$1 < q < \infty $$. Prove that there is a constant $$C(n,q) < \infty $$ such that for any 1-periodic continuous function $$g$$ on $$\mathbf{R}^{n}$$ we have

[Hint: Reduce matters to the situation where $$g =\sum _{k\in \mathbf{Z}^{n}}\chi _{k+E}$$, where E is a measurable subset of [−1∕2, 1∕2) n . Express the L q, 1 norm of $$gL_{\varepsilon /q}$$ in terms of its distribution function and for $$0 <\lambda < 1$$ estimate the measure

by $$C_{n}\big(\sqrt{n} +\big (\frac{q} {\pi \varepsilon } \log \frac{1} {\lambda } \,\big)^{1/2})^{n}\,\vert E\vert $$.]

4.3.7.

Let 0 < C ₀ < ∞. Suppose that $$\{f_{t}\}_{t\in \mathbf{R}^{+}}$$ is a family of measurable functions on a measure space X that satisfies

for every finite subset F of $$\mathbf{R}^{+}$$. (a) Suppose that for each x ∈ X, the function $$t\mapsto f_{t}(x)$$ is continuous. Show that

(b) Prove that for any t > 0 there is a measurable function $$\widetilde{f}_{t}$$ on X that is a.e. equal to f t such that

[Hint: Part (a): Notice that in view of the Lebesgue monotone convergence theorem, we have $$\big\|\sup _{t\in \mathbf{Q}}\vert f_{t}\vert \big\|_{L^{p}} \leq C_{0}$$. Also, for each $$x \in X$$ we have $$\sup _{t\in \mathbf{Q}}\vert f_{t}(x)\vert =\sup _{t\in \mathbf{R}^{+}}\vert f_{t}(x)\vert $$ by continuity. Part (b): Let $$a =\sup _{F}\|\sup _{t\in F}\vert f_{t}\vert \|_{L^{p}} \leq C_{0},$$ where the supremum is taken over all finite subsets F of $$\mathbf{R}$$. Pick an increasing sequence of finite sets F n such that $$\|\sup _{t\in F_{n}}\vert f_{t}\vert \|_{L^{p}} \rightarrow a$$ as $$n \rightarrow \infty $$. Let $$g =\sup _{n}\sup _{t\in F_{n}}\vert f_{t}\vert $$ and note that $$\|g\|_{L^{p}} = a.$$ Then for any $$s \in \mathbf{R}$$ we have

This implies $$\|\max (\vert f_{s}\vert,g)\|_{L^{p}} \leq a =\| g\|_{L^{p}}$$, so that $$\vert f_{s}\vert \leq g$$ a.e. for all $$s \in \mathbf{R}$$. ]

4.3.8. (E. Prestini) Show that for $$f \in L^{2}(\mathbf{T}^{2})$$ we have that

for almost all $$(x_{1},x_{2})$$ in $$\mathbf{T}^{2}$$.

[Hint: Use the splitting $$\widehat{f}(m_{1},m_{2}) =\widehat{ f}(m_{1},m_{2})\chi _{\vert m_{2}\vert \leq \vert m_{1}\vert ^{2}} +\widehat{ f}(m_{1},m_{2})\chi _{\vert m_{2}\vert >\vert m_{1}\vert ^{2}}$$ and apply the idea of the proof of Theorem 4.3.16.]

4.4 Applications to Geometry and Partial Differential Equations

In this section we discuss two applications of Fourier series. The first concerns a classical result in planar geometry and the other the heat equation.

4.4.1 The Isoperimetric Inequality

Suppose we are given a closed positively oriented nonself intersecting $$\mathcal{C}^{1}$$ curve C in the (x, y) plane of length L that encloses a region R of area A. The curve can be described in terms of its parametric equations x = x(t) and y = y(t), where t ∈ [0, 1]. Since the curve is closed, we have (x(0), y(0)) = (x(1), y(1)) and the $$\mathcal{C}^{1}$$ functions x(t), y(t) can be thought of as 1-periodic functions on the circle. The perimeter L of the curve is given by the equation

while the area of the region R enclosed by the curve is equal to

where we made use of Green’s theorem in the third equality above.

A $$\mathcal{C}^{1}$$ curve $$\gamma (t)$$ is regular if $$\gamma '(t)\neq 0$$ for all t. We have the following result relating the perimeter and the enclosed area of a region enclosed by a closed $$\mathcal{C}^{1}$$ curve.

Theorem 4.4.1.

Given a closed, positively oriented, nonself intersecting, regular, $$\mathcal{C}^{1}$$ planar curve of length L that encloses a region of area A, we have that

(4.4.1)

with equality holding if and only if the curve is a circle.

Proof.

Assume that the curve has parametric equations x = x(t), y = y(t), 0 ≤ t ≤ 1. We may assume that the curve has constant speed, i.e., it satisfies

for all t ∈ [0, 1]. This is achieved via the reparametrization of the curve in terms of the inverse function $$s^{-1}(t) =\gamma (t)$$ of the normalized arc length function

Since $$\vert (x'(t),y'(t))\vert \neq 0$$, $$t\mapsto s(t)$$ is a one-to-one and onto continuous map from $$[0,1]$$ to $$[0,1]$$. Then the curve $$t\mapsto (x(\gamma (t)),y(\gamma (t)))$$ has constant speed, since

So we can replace the map $$(x(t),y(t))$$ by $$(x(\gamma (t)),y(\gamma (t)))$$ which produces the same curve. Let

for t ∈ [0, 1]. Then in view of the preceding discussion, we may assume that the function f(t) = x(t) + iy(t) satisfies | f′(t) |  = L for all t ∈ [0, 1].

Under the assumption | f′(t) |  = L for all t ∈ [0, 1], we now show that (4.4.1) holds, with equality if and only if $$f(t) = c_{0}e^{2\pi it} + C_{0}$$ for some $$c_{0},C_{0} \in \mathbf{C}$$ with $$\vert c_{0}\vert = \frac{L} {2\pi }$$. To prove this claim we argue as follows:

establishing (4.4.1), but we need to explain why the inequality

(4.4.2)

is valid. Indeed, we have

where the series converges in L ². Thus we have

(4.4.3)

which proves (4.4.2).

Now suppose that equality holds in (4.4.1), then we must have equality in (4.4.2) and thus in (4.4.3), which implies that $$\widehat{f}(m) = 0$$ when | m | ≥ 2; hence for all t ∈ [0, 1] we must have

(4.4.4)

where c, c′ are complex numbers. But since$$ \|f'\|_{L^2}=L$$, it follows that

(4.4.5)

and since $$\vert f'(t)\vert = L$$ for all t ∈ [0, 1], it follows that

(4.4.6)

for all t ∈ [0, 1]. Combining (4.4.5) and (4.4.6) we obtain

(4.4.7)

Inserting t = 0 and t = 1∕8 in (4.4.7) and using that $$\mbox{ Im}\,(iz) = -\mbox{ Re}\,z$$, we deduce that $$\mbox{ Re}\,c\overline{c'} = \mbox{ Im}\,c\overline{c'} = 0$$. This implies that either c or c′ is zero. In either case (4.4.4) and (4.4.5) imply that f(t) is a circle of radius $$L/2\pi$$ centered at the point $$\widehat{f}(0)$$. □ 

4.4.2 The Heat Equation with Periodic Boundary Condition

Let k > 0 be a fixed quantity. Consider the partial differential equation

(4.4.8)

which is called the heat equation . Assume that there is an initial condition

(4.4.9)

for a given $$\mathcal{C}^{\infty }$$ function f on $$\mathbf{R}^{n}$$ which is assumed to be 1-periodic in every variable.

We would like to find a continuous function F(t, x) on $$[0,\infty ) \times \mathbf{R}^{n}$$ which is $$\mathcal{C}^{\infty }$$ on $$(0,\infty ) \times \mathbf{R}^{n}$$ such that

for all t ≥ 0 and all $$e_{j} = (0,\ldots,0,1,0,\ldots,0)$$, so that F solves the equation (4.4.8).

The function F(t, x) represents the temperature of a body at time t > 0 at the location $$(x_{1},\ldots,x_{n})$$. Since the initial temperature f is 1-periodic in each variable, we expect F(t, ⋅ ) to also be periodic in each variable. For example, F(t, x) is a good model for the temperature of the torus $$\{(e^{2\pi ix_{1}},\ldots,e^{2\pi ix_{n}}):\,\, x_{j} \in \mathbf{R}\}$$ at time t > 0, given that its temperature at time t = 0 is f(x). When n = 1, F(t, x) models the temperature of a infinitesimally thin ring, thought of as the unit circle, at time t > 0 at the location $$e^{2\pi ix}$$.

Let us suppose there is a continuous function F(t, x) on $$[0,\infty ) \times \mathbf{R}^{n}$$ which is $$\mathcal{C}^{\infty }$$ on $$(0,\infty ) \times \mathbf{R}^{n}$$ that solves the equation (4.4.8) and satisfies $$F(t,x + e_{j}) = F(t,x)$$ for all $$x \in \mathbf{R}^{n}$$ and t ≥ 0. Denote by c m (t) the Fourier coefficient of the function $$x\mapsto F(t,x)$$ defined by

Then c m (t) is a continuous function on $$[0,\infty )$$ since F is continuous in the variable t. For the same reason, c m is a smooth function on $$(0,\infty )$$ whose jth derivative is given by

for any $$j = 1,2,\ldots$$. Using equation (4.4.8) we obtain that

where the last identity is due to an integration by parts in which the boundary terms cancel each other in view of the periodicity of the integrand in x. Also $$c_{m}(0) =\widehat{ f}(m)$$. The ordinary differential equation $$c_{m}'(t) = -4\pi ^{2}\vert m\vert ^{2}k\,c_{m}(t)$$ with initial condition $$c_{m}(0) =\widehat{ f}(m)$$ is easily solved by separating the variables

(4.4.10)

yielding the solution

We may therefore define the function

(4.4.11)

on $$[0,\infty ) \times \mathbf{R}^{n}$$ and observe the following:

(a)

F is continuous on $$[0,\infty ) \times \mathbf{R}^{n}$$ and $$\mathcal{C}^{\infty }$$ on $$(0,\infty ) \times \mathbf{R}^{n}$$.

(b)

F satisfies the heat equation (4.4.8) and the initial condition (4.4.9).

(c)

F is 1-periodic in each of the last n variables.

These statements can be easily proved by passing the differentiation inside the sum, in view of the rapid convergence of the series in (4.4.11) due to the fact that the periodic function f is $$\mathcal{C}^{\infty }(\mathbf{R}^{n})$$. Furthermore, F is unique with properties (a), (b), and (c), since any other function G(x, t) with these properties is derived in the preceding way, and so it has to be equal to F(x, t).

Definition 4.4.2.

Define the heat kernel

for t > 0. Notice that the series defining H t is absolutely convergent for any t > 0. The importance of the heat kernel lies in the fact that one can express the solution F(x, t) of (4.4.8) in terms of the convolution $$F(x,t) = (f {\ast} H_{t})(x)$$.

We summarize these facts in the following proposition.

Proposition 4.4.3.

Let k > 0 be fixed and let f be in $$\mathcal{C}^{\infty }(\mathbf{R}^{n})$$ . Assume that f is 1-periodic function in each variable. Then the heat equation

(4.4.12)

under the initial condition

(4.4.13)

has a unique solution which is continuous on $$[0,\infty ) \times \mathbf{R}^{n}$$ and $$\mathcal{C}^{\infty }$$ on $$(0,\infty ) \times \mathbf{R}^{n}$$ given by

(4.4.14)

Proof.

Since f is $$\mathcal{C}^{\infty }$$, the series in (4.4.14) is rapidly convergent in m and thus it gives a continuous function on $$[0,\infty ) \times \mathbf{R}^{n}$$. Moreover, the series can be differentiated term by term in the variable t > 0, and thus it produces a $$\mathcal{C}^{\infty }$$ function on $$(0,\infty ) \times \mathbf{R}^{n}$$. By Fourier inversion (Proposition 3.​2.​5), F satisfies the initial condition (4.4.13). Finally, to verify (4.4.12), we simply notice that

where the rapid convergence of the series in m makes it possible to pass the differentiations in and out of the sum. Finally, to show uniqueness, assume that there is another solution G(t, x), continuous on $$[0,\infty ) \times \mathbf{R}^{n}$$ and $$\mathcal{C}^{\infty }$$ on $$(0,\infty ) \times \mathbf{R}^{n}$$, that can be expanded in Fourier series as follows:

Conditions (4.4.12) and the rapid decay of the coefficients c m (t) yield the ordinary differential equation (4.4.10) with initial condition $$c_{m}(0) =\widehat{ f}(m)$$, which has the solution $$c_{m}(t) =\widehat{ f}(m)e^{-4\pi ^{2}\vert m\vert ^{2}kt }$$. Thus G = F on $$[0,\infty ) \times \mathbf{R}^{n}$$. □ 

It is important to observe that the family $$\{H_{t}\}_{t>0}$$ is an approximate identity on $$\mathbf{T}^{1}$$. Indeed, the Poisson summation formula (Theorem 3.​2.​8) and the fact that the inverse Fourier transform of $$e^{-4\pi ^{2}kt\vert \xi \vert ^{2} }$$ is $$e^{-\vert x\vert ^{2}/4kt }/(2\sqrt{\pi kt})^{n}$$ [Example 2.​2.​9 and Proposition 2.​2.​11 (8)] yield that for all $$x \in [0,1]^{n}$$ we have

This identity implies that $$H_{t}(x) \geq 0$$ for all t > 0 and that

for all t > 0 and that

which tends to zero as t → 0 for any $$\delta > 0$$ in view of the Lebesgue differentiation theorem. Thus properties (i), (ii), and (iii) of approximate identities hold.

As a consequence, we have that $$\|F(t,\cdot ) - f\|_{L^{p}(\mathbf{T}^{n})} \rightarrow 0$$ as $$t \rightarrow 0$$ for $$1 \leq p < \infty $$ and F(t, ⋅ ) converges to f uniformly on $$\mathbf{T}^{n}$$; see Theorem 1.​2.​19.​

Exercises

4.4.2. Derive the following property of the heat kernel,

for all t, s > 0.

4.4.3. Consider the heat equation on $$[0,\infty ) \times \mathbf{R}$$

without a boundary condition. Show that u = 2t + x ² and $$u(t,x) = e^{-q^{2}t }e^{iqx}$$, as well as constant functions, are solutions of this equation. Prove that the set of solutions is a vector space over the field of complex numbers $$\mathbf{C}$$.

4.4.4. Suppose that a square-integrable function g(x) on $$\mathbf{R}^{n}$$ is supported in a cube $$[-A,A]^{n}$$ for some A > 0. Then we have the following representation:

where the series converges in L ²

4.4.5. This exercise provides an application of Fourier series in complex analysis. Let $$z \in \mathbf{C}\setminus \mathbf{Z}$$. Consider the function $$h_{z}(x) =\cos (2\pi zx)$$ defined on $$[-\tfrac{1} {2}, \tfrac{1} {2}]$$ extended periodically on the entire line [notice $$h_{z}(-\tfrac{1} {2}) = h_{z}(\tfrac{1} {2})$$]. (a) Compute the Fourier coefficients of h z . (b) Obtain a Fourier series expansion of h z noticing that it is a Lipschitz function. (c) Plug in x = 1∕2 to prove that

4.5 Applications to Number theory and Ergodic theory

In this section we discuss three applications of Fourier series techniques to number theory and ergodic theory.

4.5.1 Evaluation of the Riemann Zeta Function at even Natural numbers

Definition 4.5.1.

We define the Bernoulli polynomials $$\{B_{k}\}_{k=0}^{\infty }$$ on [0, 1] recursively as follows:

for $$k = 1,2,\ldots$$, and

In view of this definition we find the first few polynomials $$B_{1}(x) = x -\frac{1} {2}$$, $$B_{2}(x) = x^{2} - x + \frac{1} {6}$$, $$B_{3}(x) = x^{3} -\frac{3} {2}x^{2} + \frac{1} {2}x$$, etc. Unlike orthogonal polynomials, the Bernoulli polynomials have the remarkable property that their number of zeros in the unit interval does not increase as the degree of the polynomials increases; in fact all Bernoulli polynomials have at most three zeros in [0, 1].

Notice that for k ≥ 2 we have

thus we may think of these polynomials as functions on the circle $$\mathbf{T}^{1}$$. We extend the Bernoulli polynomials to the whole line periodically by setting $$B_{k}(x + l) = B_{k}(x)$$ for x ∈ [0, 1]. We now compute the Fourier coefficients of B k . We have

Therefore, using Corollary 3.​4.​10, we can write

where the series converges at every x ∈ (0, 1).

We have the following result concerning the Fourier expansion of the Bernoulli polynomials.

Theorem 4.5.2.

For each k ≥ 2 we have

(4.5.1)

where the series converges absolutely and uniformly on [0,1]. When k = 1 we have

(4.5.2)

for all x ∈ (0,1) and the series converges conditionally.

Proof.

We have already proved (4.5.2) and we focus attention to the case k ≥ 2. As a consequence of $$B_{k}' = kB_{k-1}$$ we obtain

Using the property that B k has integral zero over [0, 1] we evaluate the constant C k . We have

Thus

The Fourier series of B k (x) can be obtained by integrating the one for B k−1(x) for all k ≥ 2 by induction via the identity

(4.5.3)

Indeed, assume that (4.5.1) holds for some k ≥ 2. Then using (4.5.3) we obtain

Passing the limit from inside the integral to outside is allowed due to the uniform convergence of the series when k ≥ 3. In the case k = 2, one may use Exercise 3.5.6 which says that for all $$[a,b] \subseteq \mathbf{T}^{1}$$ and g integrable functions over [a, b] one has

This argument proves identity (4.5.1) for all k ≥ 2 by induction and concludes the proof. □ 

We recall the following definition from number theory.

Definition 4.5.3.

For s > 1 we define

called the Riemann zeta function .

We use the Fourier expansions of the Bernoulli polynomials to obtain the values of the Riemann zeta function for integers. When k is an even integer, identity (4.5.1) can also be written as

and inserting x = 0 yields

The polynomial B ₁(x) = x − 1∕2 has rational coefficients and thus so do all the B k by a straightforward inductive argument that uses the identity (4.5.3). Thus B k (0) is a rational number for all k ≥ 1. We conclude that

(4.5.4)

which is a rational multiple of (2π)2m , hence transcendental, since π is a transcendental number. We have therefore obtained the following.

Corollary 4.5.4.

(Euler) The value of the Riemann zeta function ζ(2m), m = 1,2,…, is equal to a rational multiple of (2π) 2m ; hence it is a transcendental number.

The corresponding statement for odd integers remains unresolved in general, as of this writing.

4.5.2 Equidistributed sequences

Here we discuss Weyl’s theorem on equidistributed sequences.

Definition 4.5.5.

A sequence $$\{a_{k}\}_{k=0}^{\infty }$$ with values in $$\mathbf{T}^{n}$$ is called equidistributed if for every cube Q in $$\mathbf{T}^{n}$$ we have

Theorem 4.5.6.

The following statements are equivalent:

(a) The sequence $$\{a_{k}\}_{k=0}^{\infty }$$ is equidistributed.

(b) For every smooth function f on $$\mathbf{T}^{n}$$ we have that

(c) For every $$m \in \mathbf{Z}^{n}\setminus \{0\}$$ we have

Proof.

We first prove the equivalence of (a) and (b). We begin by observing that (b) is a restatement of (a) if f = χ Q and Q is a cube in $$\mathbf{T}^{n}$$. Thus, if (a) holds, then (b) holds for all step functions, i.e., finite linear combinations of characteristic functions of cubes. We prove that (a) implies (b) for smooth functions. Given a smooth function f on $$\mathbf{T}^{n}$$ and given $$\varepsilon > 0$$, by the uniform continuity of f, there is a step function $$g =\sum _{ j=1}^{m}c_{j}\chi _{Q_{j}}$$ ($$c_{j} \in \mathbf{C}$$ and Q j are cubes in $$\mathbf{T}^{n}$$) such that $$\|f - g\|_{L^{\infty }} < \frac{\varepsilon } {3}$$. Since g is a finite linear combination of step functions, there is an N ₀ such that for N ≥ N ₀ we have

Since

and

it follows that for N ≥ N ₀ we have

thus (b) holds.

To prove that (b) implies (a) given a cube Q in $$\mathbf{T}^{n}$$ pick two smooth functions g and h such that

and such that g is equal to 1 on Q and vanishes off $$(1+\varepsilon )Q$$ while h is equal to 1 on $$(1-\varepsilon )Q$$ and vanishes off Q. Observe that

for some c n  > 0. Since

the sandwich theorem implies that

Since $$\varepsilon > 0$$ was arbitrary the conclusion follows.

The implication $$(b)\Rightarrow(c)$$ is trivial.

We now prove that $$(c)\Rightarrow(b)$$.

Given a smooth function f on $$\mathbf{T}^{n}$$ we write

Because of the rapid decay of the Fourier coefficients of f we can pass the limit as $$N \rightarrow \infty $$ inside the sum in m. It follows that

 □ 

Example 4.5.7.

The sequence $$\{k\sqrt{2} - [k\sqrt{2}]\}_{k=0}^{\infty }$$ is equidistributed on $$\mathbf{T}^{1}$$. We check this by verifying condition (c) of Theorem 4.5.6. Indeed if $$m \in \mathbf{Z}\setminus \{0\}$$ then

since $$m\sqrt{2}$$ is never a rational and thus the denominator never vanishes.

Naturally, the same conclusion is valid for any other irrational number in place of $$\sqrt{2}$$.

Example 4.5.8.

We examine the sequence of the first digits of powers of 2. Consider the following sequence of numbers defined for $$m = 1,2,\ldots$$

For instance we have $$d_{1} = 2,d_{2} = 4,d_{3} = 8,d_{4} = 1,d_{5} = 3,\ldots$$.

Fix an integer $$k \in \{ 1,2,3,4,5,6,7,8,9\}$$. We would like to find the frequency in which k appears as a first digit of 2 m , precisely, we would like to compute

The crucial observation is that the first digit of 2 m is equal to k if and only if there is a nonnegative integer s such that

Taking logarithms with base 10 we obtain

but since $$0 \leq \log _{10}(k)$$ and $$\log _{10}(k + 1) \leq 1$$, taking fractional parts we obtain that

and that

Since the number $$\log _{10}2$$ is irrational, it follows from Example 4.5.7 that the sequence

is equidistributed in [0, 1). Using Definition 4.5.5 in dimension n = 1 with

we obtain that

This gives the frequency in which k appears as first digit of 2 m . Notice that

as expected, and that the digit with the highest frequency that appears first in a term of the sequence $$\{1,2,4,8,16,32,64,\ldots \}$$ is 1, while the one with the lowest frequency is 9.

4.5.3 The Number of Lattice Points inside a Ball

Points in Z n are called lattice points . In this subsection we obtain the number of lattice points N(R) inside a closed ball of radius R in $$\mathbf{R}^{n}$$ centered at the origin, precisely, we compute the asymptotic behavior of

as $$R \rightarrow \infty $$. We denote by v n the volume of the closed unit ball in $$\mathbf{R}^{n}$$. We have the following result.

Theorem 4.5.9.

Let n ≥ 2. If N(R) is the number of lattice points inside the closed ball of radius R centered at zero in $$\mathbf{R}^{n}$$ , then we have that

as $$R \rightarrow \infty $$ .

Proof.

Let B be the closed unit ball in $$\mathbf{R}^{n}$$ and $$\chi _{B}$$ its characteristic function. Using the result in Appendix B.5 we have $$\widehat{\chi _{B}}(\xi ) = \frac{J_{n/2}(2\pi \vert \xi \vert )} {\vert \xi \vert ^{n/2}}$$. Now in view of the behavior of the Bessel function given in Appendix B.6 for $$\vert \xi \vert < \frac{1} {2\pi }$$ we have $$J_{n/2}(2\pi \vert \xi \vert ) \leq C\vert \xi \vert ^{\frac{n} {2} }$$. Also for $$\vert \xi \vert \geq \frac{1} {2\pi }$$ we have $$J_{n/2}(2\pi \vert \xi \vert ) \leq C\vert \xi \vert ^{-\frac{1} {2} }$$, in view of the result in Appendix B.7. Consequently, there is a constant C n such that for all $$\xi \in \mathbf{R}^{n}$$ we have

Fix a smooth nonnegative radial function ζ supported in $$\vert x\vert \leq \tfrac{1} {2}$$ with integral equal to 1 and define $$\zeta _{\varepsilon }(x) = \tfrac{1} {\varepsilon ^{n}}\zeta (\tfrac{x} {\varepsilon } )$$ for $$\varepsilon > 0$$. For $$0 <\varepsilon < \frac{1} {10}$$, define functions

These functions are even, hence their Fourier transforms are real-valued. We observe that

(4.5.5)

Indeed, we have

For $$\vert x\vert \leq 1-\varepsilon$$, $$\vert \frac{x} {\varepsilon } \vert \leq \frac{1} {\varepsilon } - 1$$, so $$\vert \frac{x} {\varepsilon } - t\vert \leq \frac{1} {\varepsilon } - 1 + \frac{1} {2} = \frac{1} {\varepsilon } -\frac{1} {2}$$ for $$\vert t\vert \leq \frac{1} {2}$$, which means

For $$\vert x\vert \geq 1,\vert \frac{x} {\varepsilon } - y\vert \geq \frac{1} {\varepsilon } -\frac{1} {\varepsilon } + \frac{1} {2} = \frac{1} {2}$$, so

proving (4.5.5). Likewise one can show that

(4.5.6)

Next we claim that

(4.5.7)

for every $$\xi \in \mathbf{R}^{n}$$ and N a large positive number. Indeed to show (4.5.7) for $$\varPhi ^{\varepsilon }$$ we write

(4.5.8)

since $$\zeta \in \mathcal{S}(\mathbf{R})$$ and $$0 <\varepsilon < \frac{1} {10}$$. The proof for $$\varPsi ^{\varepsilon }$$ is completely similar.

We now notice that for $$R > 0$$, $$m \in \mathbf{Z}^{n}\setminus \{0\}$$, and $$x \in [0,1]^{n}$$ we have

This implies that for $$\varepsilon < 1/10$$ we have

(4.5.9)

where the proof of (4.5.9) is easily deduced by considering the cases (a) $$\vert x\vert \leq R^{-1}$$ which yields a constant, (b) $$R^{-1} \leq \vert x\vert \leq (R\varepsilon )^{-1}$$ which yields a constant multiple of $$\varepsilon ^{-\frac{n-1} {2} }$$, and (c) $$(R\varepsilon )^{-1} \leq \vert x\vert $$ which also produces a constant multiple of $$\varepsilon ^{-\frac{n-1} {2} }$$ if we pick $$N > \frac{n-1} {2}$$.

Using (4.5.5) and the Poisson summation formula we write

where we used that $$(1-\varepsilon )^{n} \geq 1 - n\varepsilon$$, (4.5.8), and (4.5.9). Now pick $$\varepsilon$$ such that $$\varepsilon R^{n} =\varepsilon ^{-\frac{n-1} {2} }$$, or equivalently $$\varepsilon = R^{- \frac{2n} {n+1} }$$ to deduce the estimate

as $$R \rightarrow \infty $$.

Finally, making use of (4.5.6), and via a similar argument we write

The same choice of $$\varepsilon = R^{- \frac{2n} {n+1} }$$, yields the upper estimate for N(R).

Combining the upper and lower estimates for N(R) we obtain

as $$R \rightarrow \infty.$$ $$\square $$

Exercise 4.5.8 contains an application of Theorem 4.5.9.

Exercises

4.5.1. Prove that for all x ∈ [0, 1] we have

4.5.2. Show that for all $$z \in \mathbf{C}$$ with | z |  < 1 we have

[Hint: Use the result of Exercise 4.4.5.]

4.5.3. Suppose that a point $$x = (x_{1},\ldots,x_{n}) \in [0,1]^{n}$$ has the property that $$m \cdot x$$ is irrational for all $$m \in \mathbf{Z}^{n}\setminus \{0\}$$. Show that the sequence $$\{(kx_{1} - [kx_{1}],\ldots,kx_{n} - [kx_{n}])\}_{k=0}^{\infty }$$ is equidistributed in $$\mathbf{T}^{n}$$.

4.5.4. ([191]) Let N(x, R) be the number of lattice points inside the closed ball of radius R > 0 centered at $$x \in \mathbf{R}^{n}$$. Show that

as $$R \rightarrow \infty $$, where v n is the volume of the unit ball on $$\mathbf{R}^{n}$$.

4.5.5. (Minkowski) Let S be an open convex symmetric set in $$\mathbf{R}^{n}$$ and assume that the Fourier transform of its characteristic function satisfies the decay estimate

(This is the case if the boundary of S has nonzero Gaussian curvature.) Assume that | S |  > 2 n . Prove that S contains at least one lattice point other than the origin.

[Hint: Assume the contrary, set $$f =\chi _{\frac{1} {2} S} {\ast}\chi _{\frac{1} {2} S}$$, and apply the Poisson summation formula to f to prove that $$f(0) \geq \widehat{ f}(0)$$.]

4.5.6. For $$t \in [0,\infty )$$ let

Let $$0 = r_{0} < r_{1} < r_{2} < \cdots $$ be the sequence all of numbers r for which there exist $$m \in \mathbf{Z}^{n}$$ such that | m |  = r.

(a) Observe that N is right continuous and constant on intervals of the form $$[r_{j},r_{j+1})$$.

(b) Show that the distributional derivative of N is the measure

defined via the identity $$\langle \mu,\varphi \rangle =\sum _{ j=0}^{\infty }\#\{m \in \mathbf{Z}^{n}:\,\, \vert m\vert = r_{j}\}\,\varphi (r_{j})$$.

4.5.7. Let $$f \in \mathcal{C}^{1}((0,\infty ))$$, and let $$0 < a < b < \infty $$. Derive the identity

where N is defined in Exercise 4.5.6 and $$\int _{a}^{b}f(t)\,dN(t)$$ is the Riemann-Stieltjes integral of f with respect to N.

4.5.8. Let $$n \in \mathbf{Z}^{+}$$ and $$0 <\lambda < \infty $$.

(a) Prove that for $$k \in \mathbf{Z}^{+} \cup \{ 0\}$$ we have

for all $$0 < a < b < \infty $$ with b − a ≤ 1, where $$\omega _{n-1}$$ is the volume of $$\mathbf{S}^{n-1}$$.

(b) Show that when $$\lambda > n -\frac{n-1} {n+1}$$, the limit

exists.

(c) Prove, however, that when $$n - 1 -\frac{n-1} {n+1} <\lambda \leq n - 1$$, the limit in part (b) does not exist.

[Hint: Use Exercise 4.5.7 and Theorem 4.5.9. Part (b): For R > 1 use the identity

Notice that the main term in the first sum on the right is telescoping. Part (c): Show that $$\lim _{R\rightarrow \infty }\sum _{ \begin{array}{c}m\in \mathbf{Z}^{n}\setminus \{0\} \\ R<\vert m\vert \leq R+1\end{array}}\tfrac{e^{i\vert m\vert }} {\vert m\vert ^{\lambda }}$$ does not tend to zero.]

HISTORICAL NOTES

The boundedness of the conjugate function on the circle (Theorem 4.1.7) and, hence, the $$L^{p}$$ convergence of one-dimensional Fourier series was announced by Riesz in [292], but its proof appeared a little later in [293]. The proof of Theorem 4.1.7 in the text is attributed to S. Bochner. Luzin’s conjecture [235] on almost everywhere convergence of the Fourier series of continuous functions was announced in 1913 and settled by Carleson [54] in 1965 for the more general class of square summable functions (Theorem 4.3.14). Carleson’s theorem was later extended by Hunt [165] for the class of L p functions for all $$1 < p < \infty $$ (Theorem 4.3.15). Sjölin [325] sharpened this result by showing that the Fourier series of functions f with $$\vert f\vert (\log ^{+}\vert f\vert )(\log ^{+}\log ^{+}\vert f\vert )$$ integrable over $$\mathbf{T}^{1}$$ converge almost everywhere. Antonov [5] improved Sjölin’s result by extending it to functions f with $$\vert f\vert (\log ^{+}\vert f\vert )(\log ^{+}\log ^{+}\log ^{+}\vert f\vert )$$ integrable over $$\mathbf{T}^{1}$$. One should also consult the related results of Soria [330] and Arias de Reyna [9]. The book [10] of Arias de Reyna contains a historically motivated comprehensive study of topics related to the Carleson–Hunt theorem. Counterexamples due to Konyagin [200] show that Fourier series of functions $$f$$ with $$\vert f\vert (\log ^{+}\vert f\vert )^{\frac{1} {2} }(\log ^{+}\log ^{+}\vert f\vert )^{-\frac{1} {2} -\varepsilon }$$ integrable over $$\mathbf{T}^{1}$$ may diverge when $$\varepsilon > 0$$. Examples of continuous functions whose Fourier series diverge exactly on given sets of measure zero are given in Katznelson [189] and Kahane and Katznelson [183].

The extension of the Carleson–Hunt theorem to higher dimensions for square summability of Fourier series (Theorem 4.3.16) is a rather straightforward consequence of the one-dimensional result and was independently obtained by Fefferman [112], Sjölin [325], and Tevzadze [359]. An example showing that the circular partial sums of a Fourier series may not converge in $$L^{p}(\mathbf{T}^{n})$$ for n ≥ 2 and p ≠ 2 was obtained by Fefferman [113]. This example also shows that there exist $$L^{p}$$ functions on $$\mathbf{T}^{n}$$ for n ≥ 2 whose circular partial sums do not converge almost everywhere when $$1 \leq p < 2$$. Indeed, if the opposite happened, then the maximal operator $$f \rightarrow \sup _{N\geq 0}\vert \widetilde{D}(n,N) {\ast} f\vert $$ would have to be finite a.e. for all $$f \in L^{p}(\mathbf{T}^{n})$$, and by Stein’s theorem [335] it would have to be of weak type (p, p) for some 1 < p < 2. But this would contradict Fefferman’s counterexample on $$L^{p_1}$$ for some p < p ₁ < 2. On the other hand, almost everywhere is valid for the square partial sums of functions f with $$\vert f\vert (\log ^{+}\vert f\vert )^{n}(\log ^{+}\log ^{+}\log ^{+}\vert f\vert )$$ integrable over $$\mathbf{T}^{n}$$, as shown by Antonov [6]; see also Sjölin and Soria [327].

The development of the complex methods in the study of Fourier series was pioneered by the Russian school, especially Luzin and his students Kolmogorov, Menshov, and Privalov. The existence of an integrable function on $$\mathbf{T}^{1}$$ whose Fourier series diverges almost everywhere (Theorem 4.2.1) is due to Kolmogorov [195]. An example of an integrable function whose Fourier series diverges everywhere was also produced by Kolmogorov [198] three years later. Localization of the Bochner–Riesz means at the critical exponent $$\alpha = \tfrac{n-1} {2}$$ fails for L ¹ functions on $$\mathbf{T}^{n}$$ (see Bochner [30]) but holds for functions f such that | f | log⁺ | f | is integrable over $$\mathbf{T}^{n}$$ (see Stein [333]). The latter article also contains the $$L^{p}$$ boundedness of the maximal Bochner–Riesz operator $$\sup _{R>0}\vert B_{R}^{\alpha }(f)\vert $$ for $$1\! <\! p\! <\! \infty $$ when $$\alpha \!>\! \vert \tfrac{1} {p}\! -\!\tfrac{1} {2}\vert $$. Proposition 4.1.9 is due to Stein [331] and Theorem 4.2.5 is also due to Stein [335]. The technique that involves the points for which the set $$\{\vert x - m\vert:\,\, m \in \mathbf{Z}^{n}\}$$ is linearly independent over the rationals was introduced by Bochner [30].

Transference of regulated multipliers originated in the article of de Leeuw [94]. The methods of transference in Section 4.3 were beautifully placed into the framework of a general theory by Coifman and Weiss [70]. The key Lemma 4.3.8 is attributed to G. Weiss. Transference of maximal multipliers (Theorem 4.3.12) was first obtained by Kenig and Tomas [192] and later elaborated by Asmar, Berkson, and Gillespie [12], [13].

Paraphrasing Pappus of Alexandria, bees know than a hexagon will hold more honey than a triangle or square of the same length, but people claim a greater share of wisdom knowing that the circle of a given length holds the maximum area among all geometric shapes of equal perimeter. This reflection captures the isoperimetric inequality, which was first recorded by Pappus in the fourth century A.D. and was credited it to Zenodorus (second century B.C.). Archimedes also studied the problem, but his work on the subject, like the original writings of Zenodorus, has been lost. Rigorous modern-day proofs of this inequality can be traced to J. Steiner, K. Weierstrass, and F. Edler, whose methods are based in geometry and calculus. The proof in the text is due to A. Hurwitz. On the history of the isoperimetric inequality see [322].

The mean square error for lattice points (Exercise 4.5.4) is due to Kendall [191] while the more delicate pointwise asymptotic formula of Theorem 4.5.9 was obtained by Landau [212]. Using Landau’s formula Pinsky, Stanton, and Trapa [284] showed that the spherical partial sums of the Fourier series of the characteristic function of a sufficiently small ball in $$\mathbf{T}^{n}$$ converge at the center of the ball if and only if the dimension n is strictly less than three; this property is valid for the characteristic function of any ball as shown in Pinsky [283].

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© Springer Science+Business Media New York 2014

Loukas GrafakosClassical Fourier AnalysisGraduate Texts in Mathematics24910.1007/978-1-4939-1194-3_5

5. Singular Integrals of Convolution Type

Loukas Grafakos1 

(1)

Department of Mathematics, University of Missouri, Columbia, MO, USA

The topic of singular integrals is motivated by its intimate connection with some of the most important problems in Fourier analysis, such as that of the convergence of Fourier series. As we have seen, the L p boundedness of the conjugate function on the circle is equivalent to the L p convergence of Fourier series of L p functions. And since the Hilbert transform on the real line provides an analogue of the conjugate function on the circle, it is deeply connected with the L p convergence of Fourier integrals. It also appears in the theory of harmonic functions on the upper half space and has so many remarkable properties that deserve a careful investigation. The Hilbert transform is the prototype of all singular integrals and provides inspiration for subsequent development of the subject.

Historically, the theory of the Hilbert transform depended on techniques of complex analysis. With the development of the Calderón–Zygmund school, and the extension of one-dimensional theory to higher dimensions, real-variable methods slowly replaced complex analysis. The higher-dimensional framework proved to be flexible enough for generalizations and led to the introduction of singular integrals in other areas of mathematics. Singular integrals are nowadays intimately connected with partial differential equations, operator theory, several complex variables, and other fields. In this chapter we study singular integrals given by convolution with tempered distributions. We call such operators singular integrals of convolution type.

5.1 The Hilbert Transform and the Riesz Transforms

We begin the investigation of singular integrals with a careful study of the Hilbert transform which provides inspiration for the subsequent development of the theory.

5.1.1 Definition and Basic Properties of the Hilbert Transform

There are several equivalent ways to introduce the Hilbert transform; in this exposition we first define it as a convolution operator with a certain principal value distribution, but we later discuss other equivalent definitions.

We begin by defining a distribution W ₀ in $$\mathcal{S}'(\mathbf{R})$$ as follows:

(5.1.1)

for $$\varphi$$ in $$\mathcal{S}(\mathbf{R})$$. The function 1∕x integrated over $$[-1,-\varepsilon ]\bigcup [\varepsilon,1]$$ has mean value zero, and we may replace $$\varphi (x)$$ by $$\varphi (x) -\varphi (0)$$ in the first integral in (5.1.1). Since $$(\varphi (x) -\varphi (0))x^{-1}$$ is controlled by $$\|\varphi '\|_{L^{\infty }}$$, it follows that the limit in (5.1.1) exists. To see that W ₀ is indeed in $$\mathcal{S}'(\mathbf{R})$$, we note that the estimate

(5.1.2)

is valid. This says that $$W_{0} \in \mathcal{S}'(\mathbf{R})$$.

Definition 5.1.1.

The truncated Hilbert transform (at height $$\varepsilon$$) of a function f in L p (R), 1 ≤ p < ∞, is defined by

(5.1.3)

The Hilbert transform of $$\varphi \in \mathcal{S}(\mathbf{R})$$ is defined by

(5.1.4)

Observe that $$H^{(\varepsilon )}(f)$$ is well defined for all f ∈ L p , 1 ≤ p < ∞. This follows from Hölder’s inequality, since 1∕x is integrable to the power p′ on the set $$\vert x\vert \geq \varepsilon$$.

For Schwartz functions $$\varphi$$, the integral

may not converge absolutely for any real number x, but is defined as a limit of the absolutely convergent integrals

as $$\varepsilon \rightarrow 0$$. Such limits are called principal value integrals and are denoted by the letters p.v. Using this notation, the Hilbert transform of a Schwartz function $$\varphi$$ is

(5.1.5)

Remark 5.1.2.

We extend the definition of the Hilbert transform to a bigger class of functions. Suppose that f is an integrable function on R that satisfies a Hölder condition near every point x; that is, for any x ∈ R there are C x  > 0 and $$\varepsilon _{x} > 0$$ such that

whenever | y − x |  < δ x . Then we write

Both integrals converge absolutely; hence the limit of $$H^{(\varepsilon )}(f)(x)$$ exists as $$\varepsilon \rightarrow 0$$.

Fig. 5.1

The graph of the function H(χ E ) when E is a union of three disjoint intervals J ₁ ∪ J ₂ ∪ J ₃.

Example 5.1.3.

For the characteristic function χ [a, b] of an interval [a, b] we show that

(5.1.6)

Let us verify this identity. Pick $$\varepsilon <\min (\vert x - a\vert,\vert x - b\vert )$$. To show (5.1.6) consider the three cases 0 < x − b, x − a < 0, and x − b < 0 < x − a. In the first two cases, (5.1.6) follows immediately. In the third case we have

(5.1.7)

which yields (5.1.6). Observe that the cancellation of $$\varepsilon$$ in (5.1.7) reflects the fact that 1∕x has integral zero on symmetric intervals $$\varepsilon < \vert x\vert < c$$. Note that H(χ [a, b])(x) blows up logarithmically in x near the points a and b and decays like | x | ⁻¹ as x → ∞. See Figure 5.1.

Example 5.1.4.

Let log⁺ x = logx when x ≥ 1 and zero otherwise. Observe that the calculation in the previous example actually gives

We now give an alternative characterization of the Hilbert transform using the Fourier transform. To achieve this we need to compute the Fourier transform of the distribution W ₀ defined in (5.1.1). Fix a Schwartz function $$\varphi$$ on R. Then

(5.1.8)

(5.1.9)

Here we used the signum function

(5.1.10)

Using the results (a) and (b) in Exercise 5.1.1 we obtain that the integrals inside the square brackets in (5.1.9) are uniformly bounded by 8 and converge to $$2 \frac{\pi }{2} =\pi$$ as $$\varepsilon \rightarrow 0$$, whenever x ≠ 0. These observations make possible the use of the Lebesgue dominated convergence theorem that allows the passage of the limit inside the integral in (5.1.9). We obtain that

(5.1.11)

This implies that

(5.1.12)

In particular, identity (5.1.12) says that $$\widehat{W_{0}}$$ is a (bounded) function.

We now use identity (5.1.12) to write

(5.1.13)

This formula can be used to give an alternative definition of the Hilbert transform. An immediate consequence of (5.1.13) is that

(5.1.14)

that is, H is an isometry on L ²(R). Moreover, H satisfies

(5.1.15)

where I is the identity operator. Equation (5.1.15) is a simple consequence of the fact that $$(-i\,\mbox{ sgn}\,\xi )^{2} = -1$$. The adjoint operator H ∗ of H is uniquely defined via the identity

and we can easily obtain that H ∗ has multiplier $$\overline{ - i\,\mbox{ sgn}\,\xi } = i\,\mbox{ sgn}\,\xi$$. We conclude that H ∗ = −H. Likewise, we obtain H t  = −H.

5.1.2 Connections with Analytic Functions

We now investigate connections of the Hilbert transform with the Poisson kernel. Recall the definition of the Poisson kernel P y given in Example 1.​2.​17.​ Then for a real-valued function f in L p (R), 1 ≤ p < ∞, we have

(5.1.16)

and the integral in (5.1.16) converges absolutely by Hölder’s inequality, since the function $$t\mapsto ((x - t)^{2} + y^{2})^{-1}$$ is in L p′(R) whenever y > 0.

Let $$\mbox{ Re}\,\,z$$ and Im z denote the real and imaginary parts of a complex number z. Observe that

where z = x + i y. The function

defined on

is analytic, since its $$\partial /\partial \overline{z}$$ derivative is zero. The real part of F f (x + i y) is (P y ∗ f)(x). The imaginary part of F f (x + i y) is

where Q y is called the conjugate Poisson kernel and is given by

(5.1.17)

The function u f + i v f is analytic and thus u f (x + i y) = (f ∗ P y )(x) and v f (x + i y) = (f ∗ Q y )(x) are conjugate harmonic functions . Since the family P y , y > 0, is an approximate identity, it follows from Theorem 1.​2.​19 that P y ∗ f → f in L p (R) as y → 0. The following question therefore arises: What is the limit of f ∗ Q y as y → 0? The next result addresses this question.

Theorem 5.1.5.

Let 1 ≤ p < ∞. For any f ∈ L p ( R ) we have

(5.1.18)

in L p and almost everywhere as $$\varepsilon \rightarrow 0$$ . Moreover, for $$\varphi$$ in $$\mathcal{S}(\mathbf{R})$$ we have

(5.1.19)

as y → 0+ for all x ∈ R .

Proof.

We see that

where $$\psi _{\varepsilon }(x) =\varepsilon ^{-1}\psi (\varepsilon ^{-1}x)$$ and

(5.1.20)

Note that ψ is integrable over the line and has integral zero. Furthermore, the integrable function

(5.1.21)

is a radially decreasing majorant of ψ, i.e., it is even, decreasing on [0, ∞), and satisfies | ψ | ≤ Ψ. It follows from Theorem 1.​2.​21 (with a = 0) that $$f {\ast}\psi _{\varepsilon }\rightarrow 0$$ in L p . Also Corollary 2.​1.​19 (with a = 0) implies that $$f {\ast}\psi _{\varepsilon }\rightarrow 0$$ almost everywhere as $$\varepsilon \rightarrow 0$$.

Assertion (5.1.19) is a consequence of (5.1.18), the discussion preceding Theorem 5.1.5, and the observation that $$H^{(\varepsilon )}(\varphi )$$ converges to $$H(\varphi )$$ pointwise everywhere as $$\varepsilon \rightarrow 0$$. □ 

Remark 5.1.6.

We will show later that for f ∈ L p (R), 1 ≤ p < ∞, the expressions $$H^{(\varepsilon )}(f)$$ converge a.e. (and also in L p when p > 1) to a function $$\widetilde{H}(f)$$. This will be a consequence of Theorem 5.1.12 (or Corollary 5.3.6 when p = 1), combined with Theorem 2.​1.​14 and the observation that for Schwartz functions $$\varphi$$, $$H^{(\varepsilon )}(\varphi )$$ converge to $$H(\varphi )$$ as $$\varepsilon \rightarrow 0$$. The linear operator $$\widetilde{H}$$ defined in this way extends the Hilbert transform H initially defined on Schwartz functions and will still be denoted by H. Thus for $$f \in L^{p}(\mathbf{R})$$, $$1 \leq p < \infty $$, one has

This convergence is also valid in L p in view of the preceding observations and Theorem 5.1.5.

5.1.3 L p Boundedness of the Hilbert Transform

As a consequence of the result in Exercise 5.1.4 and of the fact that

we obtain that

(5.1.22)

for all subsets E of the real line of finite measure. Theorem 1.​4.​19 with p ₀ = q ₀ = 1 and p ₁ = q ₁ = 2 now implies that H is bounded on L p for 1 < p < 2. Duality gives that H ∗ = −H is bounded on L p for 2 < p < ∞ and hence so is H.

We give another proof of the boundedness of the Hilbert transform H on L p (R), which has the advantage that it gives the best possible constant in the resulting norm inequality when p is a power of 2.

Theorem 5.1.7.

For all 1 < p < ∞, there exists a positive constant C p such that

for all f in $$\mathcal{S}(\mathbf{R})$$ . Moreover, the constant C p satisfies C p ≤ 2p for 2 ≤ p < ∞ and C p ≤ 2p∕(p − 1) for 1 < p ≤ 2. Therefore, the Hilbert transform H admits an extension to a bounded operator on L p ( R ) when 1 < p < ∞.

Proof.

The proof we give is based on the interesting identity

(5.1.23)

which is valid whenever f is a real-valued Schwartz function. We prove (5.1.23) in two different ways. First we consider the analytic function

defined on the upper half space. We compute its square. Fix z ∈ C with $$\mbox{ Re}\,z > 0$$ and f a real-valued Schwartz function. Then for $$\varepsilon > 0$$ we have

Letting $$\varepsilon \rightarrow 0$$ and passing the limit inside the integral by the Lebesgue dominated convergence theorem, we deduce

(5.1.24)

We now let $$\mbox{ Im}\,z \rightarrow 0+$$ in (5.1.24) and use (5.1.19) in Theorem 5.1.5. We obtain

and equating the real parts we deduce (5.1.23).

To give an alternative proof of (5.1.23) we take Fourier transforms. Let

be the symbol of the Hilbert transform. We have

(5.1.25)

(5.1.26)

Averaging (5.1.25) and (5.1.26) we obtain

But the last displayed expression is equal to

in view of the identity

which is valid for all $$(\xi,\eta ) \in {\mathbf{R}}^{2}\setminus \{(0,0)\}$$ for the function $$m(\xi ) = -i\,\mbox{ sgn}\,\xi$$.

Having established (5.1.23), we can easily obtain L p bounds for H when p = 2 k is a power of 2. We already know that H is bounded on L p with norm one when p = 2 k and k = 1. Suppose that H is bounded on L p with bound c p for p = 2 k for some k ∈ Z ⁺. Then for a nonzero real-valued function f in $$\mathcal{C}_{0}^{\infty }$$ we have

Since $$\|H(f)\|_{{L}^{2p}} < \infty $$, we obtain that

If follows that

and from this we conclude that H is bounded on L 2p with bound

(5.1.27)

This completes the induction. We have proved that H maps L p to L p when p = 2 k , $$k = 1,2,\ldots$$. Interpolation now gives that H maps L p to L p for all p ≥ 2. Since H ∗ = −H, duality gives that H is also bounded on L p for 1 < p ≤ 2.

The previous proof of the boundedness of the Hilbert transform provides us with some useful information about the norm of this operator on L p (R). Let us begin with the identity

valid for $$0 < x < \frac{\pi } {2}$$. If $$c_{p} \leq \cot \frac{\pi } {2p}$$, then (5.1.27) gives that

and since $$1 =\cot \frac{\pi } {4} =\cot \frac{\pi } {2\cdot 2}$$, we obtain by induction that the numbers $$\cot \frac{\pi }{2p}$$ are indeed bounds for the norm of H on L p when p = 2 k , $$k = 1,2,\ldots$$. Duality now gives that the numbers $$\cot \frac{\pi }{2p'} =\tan \frac{\pi } {2p}$$ are bounds for the norm of H on L p when $$p = \frac{2^{k}} {2^{k}-1}$$, $$k = 1,2,\ldots$$. These bounds allow us to derive good estimates for the norm $$\|H\|_{L^{p}\rightarrow L^{p}}$$ as p → 1 and p → ∞. Indeed, since $$\cot \frac{\pi }{2p} \leq p$$ when p ≥ 2, the Riesz–Thorin interpolation theorem gives that $$\|H\|_{L^{p}\rightarrow L^{p}} \leq 2p$$ for 2 ≤ p < ∞ and by duality $$\|H\|_{L^{p}\rightarrow L^{p}} \leq \frac{2p} {p-1}$$ for 1 < p ≤ 2. This completes the proof which is worth comparing with that of Theorem 4.​1.​7.​ □ 

Remark 5.1.8.

The numbers $$\cot \frac{\pi }{2p}$$ for 2 ≤ p < ∞ and $$\tan \frac{\pi }{2p}$$ for 1 < p ≤ 2 are indeed equal to the norms of the Hilbert transform H on L p (R). This requires a more delicate argument; see Exercise 5.1.12.

Remark 5.1.9.

We may wonder what happens when p = 1 or p = ∞. The Hilbert transform of χ [a, b] computed in Example 5.1.3 is easily seen to be unbounded and not integrable, since it behaves like 1∕ | x | as x → ∞. This behavior near infinity suggests that the Hilbert transform may map L ¹ to L 1, ∞ . This is indeed the case, but this will not be shown until Section 5.3.

We now introduce the maximal Hilbert transform.

Definition 5.1.10.

The maximal Hilbert transform is the operator

(5.1.28)

defined for all f in L p , 1 ≤ p < ∞. For such f, $$H^{(\varepsilon )}(f)$$ is well defined as a convergent integral by Hölder’s inequality. Hence H (∗)(f) makes sense for f ∈ L p (R), although for some values of x, H (∗)(f)(x) may be infinite.

Example 5.1.11.

Using the result of Example 5.1.4, we obtain that

(5.1.29)

We see that in general, H (∗)(f)(x) ≠ | H(f)(x) | by taking f to be the characteristic function of the union of two disjoint closed intervals.

The definition of H gives that $$H^{(\varepsilon )}(f)$$ converges pointwise to H(f) whenever f lies in $$\mathcal{C}_{0}^{\infty }(\mathbf{R})$$. If we have the estimate $$\|H^{({\ast})}(f)\|_{L^{p}} \leq C_{p}\|f\|_{L^{p}}$$ for f ∈ L p (R), Theorem 2.​1.​14 yields that $$H^{(\varepsilon )}(f)$$ converges to H(f) a.e. as $$\varepsilon \rightarrow 0$$ for any f ∈ L p . This almost everywhere limit provides a way to describe H(f) for general f ∈ L p (R). Note that Theorem 5.1.7 implies only that H has a (unique) bounded extension on L p , but it does not provide a way to describe H(f) when f is a general L p function.

The next theorem is a simple consequence of these ideas.

Theorem 5.1.12.

There exists a constant C such that for all 1 < p < ∞ we have

(5.1.30)

Moreover, for all f in L p ( R ),$$H^{(\varepsilon )}(f)$$ converges to H(f) a.e. and in L p .

Proof.

Another proof of this theorem is given in Theorem 4.2.4 in [131 in which the asserted bound is improved.

Recall the kernels $$P_{\varepsilon }$$ and $$Q_{\varepsilon }$$ defined in (5.1.16) and (5.1.17). Fix 1 < p < ∞ and suppose momentarily that

(5.1.31)

holds whenever f is an L p function. Then we have

(5.1.32)

Using the identity

(5.1.33)

where ψ is as in (5.1.20), and applying Corollary 2.​1.​12, we obtain the estimate

(5.1.34)

where Ψ is as in (5.1.21) and M is the Hardy–Littlewood maximal function. In view of (5.1.32) and (5.1.34), we obtain for $$f \in L^{p}(\mathbf{R}^{n})$$ that

(5.1.35)

It follows immediately from (5.1.35) that H (∗) is L p bounded with norm at most $$C\max \big(p,(p - 1)^{-2}\big)$$.

We now turn to the proof of (5.1.31). It suffices to prove (5.1.31) for Schwartz functions since, given $$f \in {L}^{p}$$ there is a sequence $$\phi _{j} \in \mathcal{S}$$ such that $$\|f -\phi _{j}\|_{{L}^{p}} \rightarrow 0$$ as $$j \rightarrow \infty $$ and $$P_{\varepsilon }$$, $$Q_{\varepsilon }$$ lie in $${L}^{p'}$$ Taking Fourier transforms, we see that (5.1.31) is a consequence of the identity

(5.1.36)

To prove (5.1.36) we write

(5.1.37)

(5.1.38)

and we equate (5.1.38) and (5.1.37).

The statement in the theorem about the almost everywhere convergence of $$H^{(\varepsilon )}(f)$$ to H(f) is a consequence of (5.1.30), of the fact that the alleged convergence holds for Schwartz functions, and of Theorem 2.​1.​14.​ Finally, the L p convergence follows from the almost everywhere convergence and the Lebesgue dominated convergence theorem in view of the validity of (5.1.35). □ 

5.1.4 The Riesz Transforms

We now study an n-dimensional analogue of the Hilbert transform. It turns out that there exist n operators in R n , called the Riesz transforms, with properties analogous to those of the Hilbert transform on R.

To define the Riesz transforms, we first introduce tempered distributions W j on R n , for 1 ≤ j ≤ n, as follows. For $$\varphi \in \mathcal{S}(\mathbf{R}^{n})$$, let

One should check that indeed $$W_{j} \in \mathcal{S}'(\mathbf{R}^{n})$$. Observe that the normalization of W j is similar to that of the Poisson kernel.

Definition 5.1.13.

For 1 ≤ j ≤ n, the jth Riesz transform of f is given by convolution with the distribution W j , that is,

(5.1.39)

for all $$f \in \mathcal{S}(\mathbf{R}^{n})$$. Definition 5.1.13 makes sense for any integrable function f that has the property that for all x there exist C x  > 0, $$\varepsilon _{x} > 0$$, and δ x  > 0 such that for y satisfying | y − x |  < δ x we have $$\vert f(x) - f(y)\vert \leq C_{x}\vert x - y\vert ^{\varepsilon _{x}}$$. The principal value integral in (5.1.39) is as in Definition 5.1.1.

We now give a characterization of R j using the Fourier transform. For this we need to compute the Fourier transform of W j .

Proposition 5.1.14.

The jth Riesz transform R j is given on the Fourier transform side by multiplication by the function − iξ j ∕|ξ|. That is, for any f in $$\mathcal{S}(\mathbf{R}^{n})$$ we have

(5.1.40)

Proof.

The proof is essentially a reprise of the corresponding proof for the Hilbert transform, but it involves a few technical difficulties. Fix a Schwartz function $$\varphi$$ on R n . Then for 1 ≤ j ≤ n we have

(5.1.41)

where in the penultimate equality we used the identity $$\int _{0}^{\infty }\frac{\sin t} {t}dt = \frac{\pi } {2}$$, for which we refer to Exercise 5.1.1, while in the last equality we used the identity

(5.1.42)

which needs to be established. The passage of the limit inside the integral in the previous calculation is a consequence of the Lebesgue dominated convergence theorem, which is justified from the fact that

(5.1.43)

for all $$\varepsilon > 0$$. For a proof of (5.1.43) we again refer to Exercise 5.1.1. □ 

It remains to establish (5.1.42). Let us recall that O(n) is the set of all orthogonal n × n matrices with real entries. An invertible matrix A is called orthogonal if its transpose A t is equal to its inverse A ⁻¹, that is, A A t  = A t A = I.

Lemma 5.1.15.

The following identity is valid for all ξ ∈ R n ∖{0}:

(5.1.44)

Therefore (5.1.42) holds.

Proof.

We begin with the identity

(5.1.45)

which can be proved by noting that for k ≠ j, $$\mbox{ sgn}\,(\theta _{k})$$ has a constant sign on the hemispheres θ k  > 0 and θ k  < 0, on either of which the function θ↦θ j has integral zero.

It suffices to prove (5.1.44) for a unit vector ξ. Given ξ ∈ S n−1, pick an orthogonal n × n matrix A = (a k l ) k, l such that A e j  = ξ. Then the jth column of the matrix A is the vector $$(\xi _{1},\xi _{2},\ldots,\xi _{n})^{t}$$. We have

Next, for all $$j \in \{ 1,2,\ldots,n\}$$, we compute the value of the integral

which is obviously independent of j by symmetry. In view of the result of Appendix D.2, we write

having used the expression for ω n−2 in Appendix A.3. This proves (5.1.44). The proof of the lemma and hence that of Proposition 5.1.14 is complete. □ 

Proposition 5.1.16.

The Riesz transforms satisfy

(5.1.46)

where I is the identity operator.

Proof.

Use the Fourier transform and the identity $$\sum _{j=1}^{n}(-i\xi _{j}/\vert \xi \vert )^{2} = -1$$ to obtain that $$\sum _{j=1}^{n}R_{j}^{2}(f) = -f$$ for any f in L ²(R n ). □ 

We can express the mixed derivatives of Schwartz function in terms of its Laplacian using the Riesz transforms.

Proposition 5.1.17.

For $$\varphi$$ in $$\mathcal{S}(\mathbf{R}^{n})$$ and 1 ≤ j,k ≤ n we have

(5.1.47)

for all x ∈ R n .

Proof.

We verify the claimed identity by taking Fourier transforms. We have

and taking the inverse Fourier transform, identity (5.1.47) follows. □ 

Next we discuss a use of the Riesz transforms to partial differential equations.

Example 5.1.18.

Suppose that f is a given function in L ²(R n ) and that u is a tempered distribution on R n that solves Laplace’s equation

(5.1.48)

We express all second-order derivatives of u in terms of the Riesz transforms of f.

To solve equation (5.1.48) we first show that the tempered distribution

is supported at {0}. In view of Proposition 2.​4.​1, this implies that

where P is a polynomial of n variables (that depends on j and k) and provides a way to express the mixed partials of u in terms of the Riesz transforms of f.

To verify that $$\big(\partial _{j}\partial _{k}u + R_{j}R_{k}(f)\big)\,\,\widehat{\,}\,\,$$ is supported at {0}, we fix a Schwartz function ψ whose support does not contain the origin. Then ψ vanishes in a neighborhood of zero and we can pick $$\mathcal{C}^{\infty }$$ function η which vanishes in a smaller neighborhood of zero and is equal to 1 on the support of ψ. We define

and we notice that ζ is a bounded $$\mathcal{C}^{\infty }$$ function and so are all of its derivatives; also

Taking the Fourier transform of both sides of (5.1.48) we obtain

and multiplying by ζ (which is allowed since all derivatives of ζ lie in $$L^{\infty }\cap \mathcal{C}^{\infty }$$)

It follows that for all 1 ≤ j, k ≤ n we have

and since this holds for all Schwartz functions ψ whose support does not contain the origin, it follows that $$\big(\partial _{j}\partial _{k}u + R_{j}R_{k}(f)\big)\,\,\widehat{\,}\,\,$$ is supported at {0}.

Exercises

5.1.1. (a) Show that for all 0 < a < b < ∞ we have

(b) For a > 0 define

and show that I(a) is continuous at zero. Differentiate in a and look at the behavior of I(a) as a → ∞ to obtain the identity

Deduce that $$I(0) = \frac{\pi } {2}$$ and also derive the following identity used in (5.1.10):

(c) Argue as in part (b) to prove for a ≥ 0 the identity

[Hint: Part (a): Consider the cases b ≤ 1, a ≤ 1 ≤ b, 1 ≤ a. When a ≥ 1, integrate by parts.]

5.1.2. (a) Let $$\varphi$$ be a compactly supported $$\mathcal{C}^{m+1}$$ function on R for some m in $$\mathbf{Z}^{+}\bigcup \{0\}$$. Prove that if $$\varphi ^{(m)}$$ is the mth derivative of $$\varphi$$, then

for some $$C_{m,\varphi } > 0$$.

(b) Let $$\varphi$$ be a compactly supported $$\mathcal{C}^{m+1}$$ function on R n for some $$m \in \mathbf{Z}^{+}$$. Show that

for some $$C_{n,m,\varphi } > 0$$ and all multi-indices α with | α |  = m.

(c) Let I be an interval on the line and assume that a function h is equal to 1 on the left half of I, is equal to − 1 on the right half of I, and vanishes outside I. Prove that for $$x\notin 2I$$ we have

[Hint: Use that when $$\vert t\vert \leq \tfrac{1} {2}$$ we have log(1 + t) = t + R ₁(t), where | R ₁(t) | ≤ 2 | t | ².]

5.1.3. (a) Using identity (5.1.13) one may define H(f) as an element of $$\mathcal{S}'(\mathbf{R})$$ for bounded functions f on the line whose Fourier transform vanishes in a neighborhood of the origin. Using this interpretation, prove that

(b) Show that the operators given by convolution with the smooth function sin(t)∕t and the distribution p.v. cos(t)∕t are bounded on L p (R) whenever 1 < p < ∞.

[Hint: Use that the Fourier transform of the distribution e i x is δ 1∕2π . ]

5.1.4. ([347]) Show that the distribution function of the Hilbert transform of the characteristic function of a measurable subset E of the real line of finite measure is

[Hint: First take $$E =\bigcup _{ j=1}^{N}(a_{j},b_{j})$$, where b j  < a j+1. Show that the equation H(χ E )(x) = α has exactly one root ρ j in each open interval (a j , b j ) for 1 ≤ j ≤ N and exactly one root r j in each interval (b j , a j+1) for 1 ≤ j ≤ N, (a N+1 = ∞). Then $$\vert \{x \in \mathbf{R}:\,\, H(\chi _{E})(x) >\alpha \} \vert =\sum _{ j=1}^{N}r_{j} -\sum _{j=1}^{N}\rho _{j}$$, and this can be expressed in terms of $$\sum _{j=1}^{N}a_{j}$$ and $$\sum _{j=1}^{N}b_{j}$$. Argue similarly for the set $$\{x \in \mathbf{R}:\,\, H(\chi _{E})(x) < -\alpha \}$$. For a general measurable set E, find sets E n such that each E n is a finite union of intervals and that $$\chi _{E_{n}} \rightarrow \chi _{E}$$ in L ². Then $$H(\chi _{E_{n}}) \rightarrow H(\chi _{E})$$ in measure; thus $$H(\chi _{E_{n_{ k}}}) \rightarrow H(\chi _{E})$$ a.e. for some subsequence n k . The Lebesgue dominated convergence theorem gives $$d_{H(\chi _{E_{ n_{k}}})} \rightarrow d_{H(\chi _{E})}.$$ See Figure 5.1.]

5.1.5. Let 1 ≤ p < ∞ and let T be a linear operator defined on the space of Schwartz functions that commutes with dilations, i.e., $$T(\delta ^{\lambda }f) =\delta ^{\lambda }T(f)$$ for all $$f \in \mathcal{S}(\mathbf{R}^{n})$$ and all λ > 0. (Here $$\delta ^{\lambda }(f)(x) = f(\lambda x)$$.) Suppose that there exists a constant C > 0 such that for all $$f \in \mathcal{S}(\mathbf{R}^{n})$$ with L p norm one we have

Prove that T admits a bounded extension from L p (R n ) to $$L^{p,\infty }(\mathbf{R}^{n})$$ with norm at most C 1∕p .

[Hint: Try functions of the form $$\lambda ^{-n/p}f(\lambda ^{-1}x)/\|f\|_{L^{p}}$$ with λ > 0.]

5.1.6. Let $$\varphi$$ be in $$\mathcal{S}(\mathbf{R})$$. Prove that

5.1.7. Let T α , α ∈ R, be the operator given by convolution with the distribution whose Fourier transform is the function

(a) Show that the T α ’s are isometries on L ²(R) that satisfy

(b) Express T α in terms of the identity operator and the Hilbert transform.

5.1.8. Let Q y (j) be the jth conjugate Poisson kernel of P y defined by

Prove that

Conclude that R j (P y ) = Q y (j) and that for f in L ²(R n ) we have R j (f) ∗ P y  = f ∗ Q y (j). These results are analogous to the statements $$\widehat{Q_{y}}(\xi ) = -i\,\mbox{ sgn}\,(\xi )\widehat{P_{y}}(\xi )$$, H(P y ) = Q y , and H(f) ∗ P y  = f ∗ Q y .

5.1.9. Fix n ≥ 2. Let $$f_{0},f_{1},\ldots,f_{n}$$ be in L ²(R n ) and, for 0 ≤ j ≤ n, let $$u_{j}(x,x_{0}) = (P_{x_{0}} {\ast} f_{j})(x)$$ be the Poisson integrals of f j where $$x = (x_{1},\ldots,x_{n}) \in \mathbf{R}^{n}$$ and x ₀ > 0. Show that a necessary and sufficient condition for

is that the following system of generalized Cauchy-Riemann equations holds:

5.1.10. Prove the distributional identity

Then take Fourier transforms of both sides and use Theorem 2.​4.​6 to obtain another proof of Proposition 5.1.14.

5.1.11. (a) Prove that if T is a bounded linear operator on L ²(R) that commutes with translations and dilations and anticommutes with the reflection $$f(x)\mapsto \widetilde{f}(x) = f(-x)$$, then T is a constant multiple of the Hilbert transform.

(b) Prove that if T is a bounded linear operator on L ²(R) that commutes with translations and dilations and vanishes when applied to functions whose Fourier transform is supported in [0, ∞), then T is a constant multiple of the operator $$f\mapsto \big(\widehat{f}\chi _{(-\infty,0]}\big)\,^{\vee }\,$$.

5.1.12. ([282]) Fix 1 < p ≤ 2. (a) Show that the function $$G(x,y) = \mbox{ Re}\,\,(\vert x\vert + iy)^{p}$$ is subharmonic on R ².

(b) Let u(x, y), v(x, y) be real-valued functions on R ² such that u + i v is a holomorphic function of x + i y. Prove that G(u, v) is a subharmonic function on R ².

(c) Prove that there is a constant B p such that for all a and b reals we have

(d) Prove that for f in $$\mathcal{C}_{0}^{\infty }(\mathbf{R})$$ we have

(e) Combine the results in parts (d) and (c) with a = f(x), b = H(f)(x) to obtain that

(f) To deduce that this constant is sharp, take π∕2p′ < γ < π∕2p and let $$f_{\gamma }(x) = (x + 1)^{-1}\vert x + 1\vert ^{2\gamma /\pi }\vert x - 1\vert ^{-2\gamma /\pi }\cos \gamma$$. Then

[Hint: Part (d): Let C R be the circle of radius R centered at (0, R) in R ². Use that the integral of the subharmonic function G((P y ∗ f)(x), Q y ∗ f)(x)) over C R is at least $$2\pi R\,\mbox{ Re}\,\,(\vert (P_{R} {\ast} f)(0)\vert + i(Q_{R} {\ast} f)(0))^{p}$$ and let R → ∞. Part (f): The formula for H(f γ ) is best derived by considering the restriction of the analytic function

on the real line.]

5.2 Homogeneous Singular Integrals and the Method of Rotations

So far we have introduced the Hilbert and the Riesz transforms and we have derived the L p boundedness of the former. The boundedness properties of the Riesz transforms on L p spaces are consequences of the results discussed in this section.

5.2.1 Homogeneous Singular and Maximal Singular Integrals

We introduce singular integral operators on R n that appropriately generalize the Riesz transforms on R n . Here is the setup. We fix Ω to be an integrable function of the unit sphere S n−1 with mean value zero. Observe that the kernel

(5.2.1)

is homogeneous of degree − n just like the functions x j ∕ | x |  n+1. Since K Ω is not in L ¹(R n ), convolution with K Ω cannot be defined as an operation on Schwartz functions on R n . For this reason we introduce a distribution W Ω in $$\mathcal{S}'(\mathbf{R}^{n})$$ by setting

(5.2.2)

for $$\varphi \in \mathcal{S}(\mathbf{R}^{n})$$. Using the fact that Ω has mean value zero, we can easily see that W Ω is a well defined tempered distribution on R n . Indeed, since K Ω has integral zero over all annuli centered at the origin, we have

for suitable C ₁ and C ₂, where we used (2.​2.​2) in the last estimate. Note that the distribution W Ω coincides with the function K Ω on R n ∖{0}.

The Hilbert transform and the Riesz transforms are examples of these general operators T Ω . For instance, the function $$\varOmega (\theta ) = \frac{\theta } {\pi \vert \theta \vert } = \frac{1} {\pi } \mbox{ sgn}\,\theta$$ defined on the unit sphere $$\mathbf{S}^{0} =\{ -1,1\} \subseteq \mathbf{R}$$ gives rise to the Hilbert transform, while the function

defined on $$\mathbf{S}^{n-1} \subseteq \mathbf{R}^{n}$$ gives rise to the jth Riesz transform.

Definition 5.2.1.

Let Ω be integrable on the sphere S n−1 with mean value zero. For $$0 <\varepsilon < N$$ and $$f \in \bigcup _{1\leq p<\infty }L^{p}(\mathbf{R}^{n})$$ we define the truncated singular integral

(5.2.3)

Note that for $$f \in L^{p}(\mathbf{R}^{n})$$ we have

which implies that (5.2.3) is finite a.e. and therefore well defined. We denote by T Ω the singular integral operator whose kernel is the distribution W Ω , that is,

defined for $$f \in \mathcal{S}(\mathbf{R}^{n})$$. The associated maximal singular integral is defined by

(5.2.4)

We note that if Ω is bounded, there is no need to use the upper truncations in the definition of $$T_{\varOmega }^{(\varepsilon,N)}$$ given in (5.2.3). In this case the maximal singular integrals could be defined as

(5.2.5)

where for $$f \in \bigcup _{1\leq p<\infty }L^{p}(\mathbf{R})$$, $$\varepsilon > 0$$, and x ∈ R n , $$T_{\varOmega }^{(\varepsilon )}(f)(x)$$ is defined in terms of the absolutely convergent integral

To examine the relationship between T Ω (∗) and $$T_{\varOmega }^{({\ast}{\ast})}$$ for $$\varOmega \in L^{\infty }(\mathbf{S}^{n-1})$$, notice that

(5.2.6)

Then for f ∈ L p (R n ), 1 ≤ p < ∞, we let N → ∞ on the left in (5.2.6) and we note that the limit exists in view of the absolute convergence of the integral. Then we take the supremum over $$\varepsilon > 0$$ to deduce that T Ω (∗) is pointwise bounded by T Ω (∗∗). Since $$T_{\varOmega }^{(\varepsilon,N)} = T_{\varOmega }^{(\varepsilon )} - T_{\varOmega }^{(N)}$$, it also follows that $$T_{\varOmega }^{({\ast}{\ast})} \leq 2\,T_{\varOmega }^{({\ast})}$$; thus T Ω (∗) and T Ω (∗∗) are pointwise comparable when Ω lies in $$L^{\infty }(\mathbf{S}^{n-1})$$. This is the case with the Hilbert transform, that is, H (∗∗) is comparable to H (∗); likewise with the Riesz transforms.

A certain class of multipliers can be realized as singular integral operators of the kind discussed. Recall from Proposition 2.​4.​7 that if m is homogeneous of degree 0 and infinitely differentiable on the sphere, then $$m\,^{\vee }\,$$ is given by

for some complex constant c and some smooth Ω on S n−1 with mean value zero. Therefore, all convolution operators whose multipliers are homogeneous of degree zero smooth functions on S n−1 can be realized as a constant multiple of the identity plus an operator of the form T Ω .

Example 5.2.2.

Let $$P(\xi ) =\sum _{\vert \alpha \vert =k}b_{\alpha }\xi ^{\alpha }$$ be a homogeneous polynomial of degree k in R n that vanishes only at the origin. Let α be a multi-index of order k. Then the function

(5.2.7)

is infinitely differentiable on the sphere and homogeneous of degree zero. The operator given by multiplication on the Fourier transform by m(ξ) is a constant multiple of the identity plus an operator given by convolution with a distribution of the form W Ω for some Ω in $$\mathcal{C}^{\infty }(\mathbf{S}^{n-1})$$ with mean value zero. In this section we establish the L p boundedness of such operators when Ω has appropriate smoothness on the sphere. This, in particular, implies that m(ξ) defined by (5.2.7) lies in the space $$\mathcal{M}_{p}(\mathbf{R}^{n})$$, defined in Section 2.​5, for 1 < p < ∞.

5.2.2 L ² Boundedness of Homogeneous Singular Integrals

Next we would like to compute the Fourier transform of W Ω . This provides information as to whether the operator given by convolution with K Ω is L ² bounded. We have the following result.

Proposition 5.2.3.

Let n ≥ 2 and Ω ∈ L ¹ ( S n−1 ) have mean value zero. Then the Fourier transform of W Ω is a (finite a.e.) function given by the formula

(5.2.8)

Remark 5.2.4.

We need to show that the function of ξ on the right in (5.2.8) is well defined and finite for almost all ξ in R n . Write ξ =  | ξ | ξ′ where ξ′ ∈ S n−1 and notice that

Since Ω has mean value zero, the term $$\log \frac{1} {\vert \xi \vert }$$ multiplied by Ω(θ) vanishes when integrated over the sphere.

We need to show that

(5.2.9)

for almost all ξ′ ∈ S n−1. Integrate (5.2.9) over ξ′ ∈ S n−1 and apply Fubini’s theorem to obtain

since we are assuming that n ≥ 2. (The second-to-last identity follows from the identity in Appendix D.2.) We conclude that (5.2.9) holds for almost all ξ′ ∈ S n−1.

Since the function of ξ on the right in (5.2.8) is homogeneous of degree zero, it follows that it is a locally integrable function on R n .

Before we return to the proof of Proposition 5.2.3, we discuss the following lemma:

Lemma 5.2.5.

Let a be a nonzero real number. Then for $$0 <\varepsilon < N < \infty $$ we have

(5.2.10)

(5.2.11)

(5.2.12)

(5.2.13)

Proof.

We first prove (5.2.10) and (5.2.11). By the fundamental theorem of calculus we write

and from this expression, we clearly obtain (5.2.11). But the first integral of the same expression converges to − log | a | as $$\varepsilon \rightarrow 0$$ while the second integral converges to zero as N → ∞ by an integration by parts. This proves (5.2.10).

To prove (5.2.12) and (5.2.13) we need to know that the expressions

(5.2.14)

tend to $$\tfrac{\pi }{2}$$ as $$\varepsilon \rightarrow 0$$ and N → ∞ and are bounded by 4. Both statements follow from Exercise 5.1.1. □ 

Let us now prove Proposition 5.2.3.

Proof.

Let us set ξ′ = ξ∕ | ξ | . We have the following:

where we used the Lebesgue dominated convergence theorem to pass the limit inside, Lemma 5.2.5, and Remark 5.2.4. We were able to subtract $$\cos (2\pi r\vert \xi \vert )$$ from the r integral in the previous calculation, since Ω has mean value zero over the sphere. Also, the use of the dominated convergence theorem is justified from the fact that the function

lies in L ¹(S n−1 ×R n ). Moreover, all the interchanges of integrals are well justified by Fubini’s theorem. □ 

Corollary 5.2.6.

Let Ω ∈ L ¹ ( S n−1 ) have mean value zero. Then for almost all ξ′ in S n−1 the integral

(5.2.15)

converges absolutely. Moreover, the associated operator T Ω maps L ² ( R n ) to itself if and only if

(5.2.16)

Proof.

To obtain the absolute convergence of the integral in (5.2.15) we integrate over ξ′ ∈ S n−1 and we apply Fubini’s theorem. The assertion concerning the boundedness of T Ω on L ² is an immediate consequence of Proposition 5.2.3 and Theorem 2.​5.​10.​ □ 

There exist functions Ω in L ¹(S n−1) with mean value zero such that the expressions in (5.2.16) are equal to infinity; consequently, not all such Ω give rise to bounded operators on L ²(R n ). Observe, however, that for Ω odd i.e., Ω(−θ) = −Ω(θ) for all θ ∈ S n−1, (5.2.16) trivially holds, since $$\log \frac{1} {\vert \xi \cdot \theta \vert }$$ is even and its product against an odd function must have integral zero over S n−1. We conclude that singular integrals T Ω with odd Ω are always L ² bounded.

5.2.3 The Method of Rotations

Having settled the issue of L ² boundedness for singular integrals of the form T Ω with Ω odd, we turn our attention to their L p boundedness. A simple procedure called the method of rotations plays a crucial role in the study of operators T Ω when Ω is an odd function. This method is based on the use of the directional Hilbert transforms. Fix a unit vector θ in R n . For a Schwartz function f on R n let

(5.2.17)

We call $$\mathcal{H}_{\theta }(f)$$ the directional Hilbert transform of f in the direction θ. For functions $$f \in \mathcal{S}(\mathbf{R}^{n})$$ the integral in (5.2.17) is well defined, since it converges rapidly at infinity and by subtracting the constant f(x), it also converges near zero.

Likewise, we define the directional maximal Hilbert transforms. For a function f in $$\bigcup _{1\leq p<\infty }L^{p}(\mathbf{R}^{n})$$ and $$0 <\varepsilon < N < \infty $$ we let

We observe that for any fixed $$0 <\varepsilon < N < \infty $$ and $$f \in L^{p}(\mathbf{R}^{n})$$, $$\mathcal{H}_{\theta }^{(\varepsilon,N)}(f)$$ is well defined almost everywhere. Indeed, by Minkowski’s integral inequality we obtain

which implies that $$\mathcal{H}_{\theta }^{(\varepsilon,N)}(f)(x)$$ is finite for almost all x ∈ R n . Thus $$\mathcal{H}_{\theta }^{({\ast}{\ast})}(f)$$ is well defined for f in $$\bigcup _{1\leq p<\infty }L^{p}(\mathbf{R}^{n})$$.

Theorem 5.2.7.

If Ω is odd and integrable over S n−1 , then T Ω and T Ω (∗∗) are L p bounded for all 1 < p < ∞. More precisely, T Ω initially defined on Schwartz functions has a bounded extension on $$L^{p}(\mathbf{R}^{n})$$ (which is also denoted by T Ω ).

Proof.

Let e j be the usual unit vectors in S n−1. The operator $$\mathcal{H}_{e_{1}}$$ is obtained by applying the Hilbert transform in the first variable followed by the identity operator in the remaining variables. Clearly, $$\mathcal{H}_{e_{1}}$$ is bounded on $$L^{p}(\mathbf{R}^{n})$$ with norm equal to that of the Hilbert transform on L p (R). Next observe that the following identity is valid for all matrices A ∈ O(n):

(5.2.18)

This implies that the L p boundedness of $$\mathcal{H}_{\theta }$$ can be reduced to that of $$\mathcal{H}_{e_{1}}$$. We conclude that $$\mathcal{H}_{\theta }$$ is L p bounded for 1 < p < ∞ with norm bounded by the norm of the Hilbert transform on L p (R) for every θ ∈ S n−1.

Identity (5.2.18) is also valid for $$\mathcal{H}_{\theta }^{(\varepsilon,N)}$$ and $$\mathcal{H}_{\theta }^{({\ast}{\ast})}$$. Consequently, $$\mathcal{H}_{\theta }^{({\ast}{\ast})}$$ is bounded on L p (R n ) for 1 < p < ∞ with norm at most that of H (∗∗) on L p (R).

Next we realize a general singular integral T Ω with Ω odd as an average of the directional Hilbert transforms $$\mathcal{H}_{\theta }$$. We start with f in $$\bigcup _{1\leq p<\infty }L^{p}(\mathbf{R}^{n})$$ and the following identities:

where the first follows by switching to polar coordinates and the second one is a consequence of the first one and the fact that Ω is odd via the change variables θ↦ −θ. Averaging the two identities, we obtain

(5.2.19)

It follows from the identity in (5.2.19) that

(5.2.20)

from which we conclude that

(5.2.21)

Using the Lebesgue dominated convergence theorem, we see that for f in $$\mathcal{S}(\mathbf{R}^{n})$$, we can pass the limits as $$\varepsilon \rightarrow 0$$ and $$N \rightarrow \infty $$ inside the integral in (5.2.20), concluding that

(5.2.22)

for $$f \in \mathcal{S}(\mathbf{R}^{n})$$. The L p boundedness of T Ω and T Ω (∗∗) for Ω odd are then trivial consequences of (5.2.22) and (5.2.21) via Minkowski’s integral inequality. □ 

Corollary 5.2.8.

The Riesz transforms R j and the maximal Riesz transforms R j (∗) are bounded on L p ( R n ) for 1 < p < ∞.

Proof.

The assertion follows from the fact that the Riesz transforms have odd kernels. Since the kernel of R j decays like | x | −n near infinity, it follows that R j (∗)(f) is well defined for $$f \in L^{p}(\mathbf{R}^{n})$$. Since R j (∗) is pointwise bounded by 2R j (∗∗), the conclusion follows from Theorem 5.2.7. □ 

Remark 5.2.9.

It follows from the proof of Theorem 5.2.7 and from Theorems 5.1.7 and 5.1.12 that whenever Ω is an odd function on S n−1, we have

for some a > 0 independent of p and the dimension.

5.2.4 Singular Integrals with Even Kernels

Since a general integrable function Ω on S n−1 with mean value zero can be written as a sum of an odd and an even function, it suffices to study singular integral operators T Ω with even kernels. For the rest of this section, fix an integrable even function Ω on S n−1 with mean value zero. The following idea is fundamental in the study of such singular integrals. Proposition 5.1.16 implies that

(5.2.23)

If R j T Ω were another singular integral operator of the form $$T_{\varOmega _{j}}$$ for some odd Ω j , then the boundedness of T Ω would follow from that of $$T_{\varOmega _{j}}$$ via the identity (5.2.23) and Theorem 5.2.7. It turns out that R j T Ω does have an odd kernel, but it may not be integrable on S n−1 unless Ω itself possesses an additional amount of integrability. The amount of extra integrability needed is logarithmic, more precisely of this sort:

(5.2.24)

Observe that

which says that the norm $$\|\varOmega \|_{L^{1}}$$ is always controlled by a dimensional constant multiple of c Ω + 1. The following theorem is the main result of this section.

Theorem 5.2.10.

Let n ≥ 2 and let Ω be an even integrable function on S n−1 with mean value zero that satisfies (5.2.24) . Then the corresponding singular integral T Ω is bounded on $$L^{p}(\mathbf{R}^{n})$$ , 1 < p < ∞, with norm at most a dimensional constant multiple of the quantity $$\max \big((p - 1)^{-2},p^{2}\big)(c_{\varOmega } + 1)$$ .

If the operator T Ω in Theorem 5.2.10 is weak type (1, 1), then the estimate on the L p operator norm of T Ω can be improved to $$\|T_{\varOmega }\|_{L^{p}\rightarrow L^{p}} \leq C_{n}(p - 1)^{-1}$$ as p → 1. This is indeed the case; see the historical comments at the end of this chapter.

Proof.

Let W Ω be the distributional kernel of T Ω . We have that W Ω coincides with the function Ω(x∕ | x | ) | x | −n on R n ∖{0}. Using Proposition 5.2.3 and the fact that Ω is an even function, we obtain the formula

(5.2.25)

which implies that $$\widehat{W_{\varOmega }}$$ is itself an even function. Now, using Exercise 5.2.3 and condition (5.2.24), we conclude that $$\widehat{W_{\varOmega }}$$ is a bounded function. Therefore, T Ω is L ² bounded. To obtain the L p boundedness of T Ω , we use the idea mentioned earlier involving the Riesz transforms. In view of (5.1.46), we have that

(5.2.26)

where T j  = R j T Ω . Equality (5.2.26) makes sense as an operator identity on L ²(R n ), since T Ω and each R j are well defined and bounded on L ²(R n ).

The kernel of the operator T j is the inverse Fourier transform of the distribution $$-i\tfrac{\xi _{j}} {\vert \xi \vert }\widehat{W_{\varOmega }}(\xi )$$, which we denote by K j . At this point we know only that K j is a tempered distribution whose Fourier transform is the function $$-i\tfrac{\xi _{j}} {\vert \xi \vert }\widehat{W_{\varOmega }}(\xi )$$. Our first goal is to show that K j coincides with an integrable function on an annulus. To prove this assertion we write

where W Ω ⁰ is a distribution and $$W_{\varOmega }^{1},W_{\varOmega }^{\infty }$$ are functions defined by

We now fix a $$j \in \{ 1,2,\ldots,n\}$$ and we write

where

Notice that K j ⁰ is well defined via Theorem 2.​3.​21.​

Define the annulus

For a smooth function ϕ supported in the annulus 2∕3 <  | x |  < 3∕2 we have

where the action of the distribution W Ω ⁰ on R j (ϕ) is justified by fact that R j (ϕ)(y) is smooth on the support of W Ω ⁰; note | x − y | ≥ 1∕6. Moreover, $$\langle \widehat{W_{\varOmega }^{0}}(\xi ),-i\tfrac{\xi _{j}} {\vert \xi \vert }\phi \,^{\vee }\,(\xi )\rangle$$ should be interpreted as a convergent integral.

It follows that for x ∈ A, the absolute value of the convolution of W Ω ⁰ with the kernel of the Riesz transform R j is

(5.2.27)

(5.2.28)

where we used the fact that Ω(y∕ | y | ) | y | −n has integral zero over annuli of the form $$\varepsilon < \vert y\vert < \frac{1} {2}$$, the mean value theorem applied to the function x j  | x | −(n+1), and the fact that | x − y | ≥ 1∕6 for x in the annulus A. We conclude that on A, K j ⁰ coincides with the bounded function inside the absolute value in (5.2.27).

Likewise, for x ∈ A we have

(5.2.29)

from which it follows that on the annulus A, K j ∞ coincides with the bounded function inside the absolute value in (5.2.29) or in (5.2.28).

Now observe that condition (5.2.24) gives that the function W Ω ¹ satisfies

Since the Riesz transform R j is countably subadditive and maps L p to L p with norm at most 4(p − 1)⁻¹ for 1 < p < 2, it follows from Exercise 1.3.7 that $$K_{j}^{1} = R_{j}(W_{\varOmega }^{1})$$ is integrable over the ball | x | ≤ 3∕2 and moreover, it satisfies

Furthermore, since $$\widehat{K_{j}}$$ is homogeneous of degree zero, K j is a homogeneous distribution of degree − n (Exercise 2.3.9). This means that for all test functions $$\varphi$$ and all λ > 0 we have

(5.2.30)

where $$\delta ^{\lambda }(\varphi )(x) =\varphi (\lambda x)$$. But for $$\varphi \in \mathcal{C}_{0}^{\infty }$$ supported in the annulus 3∕4 <  | x |  < 4∕3 and for λ in (8∕9, 9∕8) we have that $$\delta ^{\lambda ^{-1} }(\varphi )$$ is supported in A and thus we can express (5.2.30) as convergent integrals as follows:

(5.2.31)

From this it would be ideal to be able to directly obtain that $$K_{j}(x) =\lambda ^{n}K_{j}(\lambda x)$$ for all 8∕9 <  | x |  < 9∕8 and 8∕9 < λ < 9∕8, in particular when λ =  | x | ⁻¹. But unfortunately, we can only deduce that for every λ ∈ (8∕9, 9∕8), K j (x) = λ n K j (λ x) holds for all x in the annulus except a set of measure zero that depends on λ. To be able to define the restriction of K j on S n−1, we employ a more delicate argument.

For any J subinterval of [8∕9, 9∕8] we obtain from (5.2.31) that

where integral with the slashed integral denotes the average of a function over the set J. Since $$\varphi$$ was an arbitrary $$\mathcal{C}_{0}^{\infty }$$ function supported in the annulus 3∕4 <  | x |  < 4∕3, it follows that for every J subinterval of [8∕9, 9∕8], there is a null subset E J of the annulus A′ = { x:  27∕32 <  | x |  < 32∕27} such that

(5.2.32)

for all x ∈ A′∖ E J .

Let $$J_{0} = [\sqrt{8/9},\sqrt{9/8}]$$. We claim that there is a set of null subset E of A′ such that for all x ∈ A′∖ E we have

(5.2.33)

for every r in J ₀. Indeed, let E be the union of $$E_{rJ_{0}}$$ over all r in J ₀ ∩Q. Then in view of (5.2.32), identity (5.2.33) holds for x ∈ A′∖ E and J ₀ ∩Q. But for a fixed x in A′∖ E, the function of r on the right hand side of (5.2.33) is constant on the rationals and is also continuous (in r), hence it must be constant for all r ∈ J ₀. Thus the claim follows since both sides of (5.2.33) are equal to (5.2.32).

Writing x = δ θ, where 27∕32 < δ < 32∕27 and θ ∈ S n−1, it follows by Fubini’s theorem that there is a δ ∈ (27∕32, 32∕27) (in fact almost all δ have this property) such that

(5.2.34)

for almost all θ ∈ S n−1 and all r ∈ J ₀. We fix such a δ, which we denote δ ₀.

We now define a function Ω j on S n−1 by setting

for all r ∈ J ₀. The function Ω j is defined almost everywhere and is integrable over S n−1, since K j is integrable over the annulus A.

Let $$e_{1} = (1,0,\ldots,0)$$. Let Ψ be a $$\mathcal{C}_{0}^{\infty }(\mathbf{R}^{n})$$ nonzero, nonnegative, radial, and supported in the annulus $$32/(27\sqrt{2}) < \vert x\vert < 27\sqrt{2}/32$$ around S n−1. We start with

which holds for all r ∈ J ₀, we multiply by Ψ(r e ₁), and we integrate over S n−1 and over (0, ∞) with respect to the measure d r∕r. We obtain

in view of the homogeneity of K j . But also, for some constant c Ψ ′ we have

since by (5.2.25), $$\frac{-i\xi _{j}} {\vert \xi \vert } \widehat{W_{\varOmega }}(\xi )$$ is an odd function. We conclude that Ω j has mean value zero over S n−1.

Thus Ω ∈ L ¹(S n−1) has mean value zero and the distribution $$W_{\varOmega _{j}}$$ is well defined. We claim that

(5.2.35)

To establish (5.2.35), we show first that $$\big\langle K_{j},\varphi \big\rangle =\big\langle W_{\varOmega _{j}},\varphi \big\rangle$$ whenever $$\varphi$$ is supported in the annulus 8∕9 <  | x |  < 9∕8. Using (5.2.32) we have

having used (5.2.34) in the second to last equality.

Given a general $$\mathcal{C}_{0}^{\infty }$$ function $$\varphi$$ whose support is contained in an annulus of the form M ⁻¹ <  | x |  < M, for some M > 0, via a smooth partition of unity, we write $$\varphi$$ as a finite sum of smooth functions $$\varphi _{k}$$ whose supports are contained in annuli of the form 8s∕9 <  | x |  < 9s∕8 for some s > 0. These annuli can be brought inside the annulus 8∕9 <  | x |  < 9∕8 by a dilation. Since both K j and $$W_{\varOmega _{j}}$$ are homogeneous distributions of degree − n and agree on the annulus 8∕9 <  | x |  < 9∕8 they must agree on annuli 8s∕9 <  | x |  < 9s∕8. Consequently, $$\langle K_{j},\varphi \rangle =\langle W_{\varOmega _{j}},\varphi \rangle$$ for all $$\varphi \in \mathcal{C}_{0}^{\infty }(\mathbf{R}^{n}\setminus \{0\})$$. Therefore, $$K_{j} - W_{\varOmega _{j}}$$ is supported at the origin, and since it is homogeneous of degree − n, it must be equal to b δ ₀, a constant multiple of the Dirac mass. But $$\widehat{K_{j}}$$ is an odd function and hence K j is also odd. It follows that $$W_{\varOmega _{j}}$$ is an odd function on R n ∖{0}, which implies that Ω j is an odd function. We say that $$u \in \mathcal{S}'(\mathbf{R}^{n})$$ is odd if $$\widetilde{u} = -u$$, where $$\widetilde{u}$$ is defined by $$\langle \widetilde{u},\psi \rangle =\langle u,\widetilde{\psi }\rangle$$ for all $$\psi \in \mathcal{S}(\mathbf{R}^{n})$$ and $$\widetilde{\psi }(x) =\psi (-x)$$. We have that $$K_{j} - W_{\varOmega _{j}}$$ is an odd distribution, and thus b δ ₀ must be an odd distribution. But if b δ ₀ is odd, then b = 0. We conclude that for each j there exists an odd integrable function Ω j on S n−1 with $$\|\varOmega _{j}\|_{L^{1}}$$ controlled by a constant multiple of c Ω + 1 such that (5.2.35) holds.

Then we use (5.2.26) and (5.2.35) to write

and appealing to the boundedness of each $$T_{\varOmega _{j}}$$ (Theorem 5.2.7) and to that of the Riesz transforms, we obtain the required L p boundedness for T Ω . □ 

We note that Theorem 5.2.10 holds for all Ω ∈ L ¹(S n−1) that satisfy (5.2.24), not necessarily even Ω. Simply write Ω = Ω e +Ω o  , where Ω e is even and Ω o is odd, and check that condition (5.2.24) holds for Ω e .

5.2.5 Maximal Singular Integrals with Even Kernels

We have the corresponding theorem for maximal singular integrals.

Theorem 5.2.11.

Let Ω be an even integrable function on S n−1 with mean value zero that satisfies (5.2.24) . Then the corresponding maximal singular integral T Ω (∗∗) , defined in (5.2.4) , is bounded on $$L^{p}(\mathbf{R}^{n})$$ for 1 < p < ∞ with norm at most a dimensional constant multiple of $$\max (p^{2},(p - 1)^{-2})(c_{\varOmega } + 1)$$ .

Proof.

For $$f \in L_{\text{loc}}^{1}(\mathbf{R}^{n})$$, x define the maximal function of f in the direction θ by setting

(5.2.36)

In view of Exercise 5.2.5 we have that M θ is bounded on $$L^{p}(\mathbf{R}^{n})$$ with norm at most 3 p (p − 1)⁻¹.

Fix Φ a smooth radial function such that Φ(x) = 0 for | x | ≤ 1∕4, Φ(x) = 1 for | x | ≥ 3∕4, and 0 ≤ Φ(x) ≤ 1 for all x in R n . For $$f \in L^{p}(\mathbf{R}^{n})$$ and $$0 <\varepsilon < N < \infty $$ we introduce the smoothly truncated singular integral

and the corresponding maximal singular integral operator

(5.2.37)

Computing the supremum in (5.2.37), we first consider the case where $$N > 4\varepsilon$$.

For f in $$L^{p}(\mathbf{R}^{n})$$ (for some 1 < p < ∞), we have

Now if $$N \leq 4\varepsilon$$, then the function $$\varPhi \left (\tfrac{y} {\varepsilon } \right ) -\varPhi \left ( \tfrac{y} {N}\right ) -\chi _{\varepsilon \leq \vert y\vert \leq N}$$ is bounded by 3 and is supported in the annulus $$\frac{\varepsilon }{4} \leq \vert y\vert \leq 4\varepsilon$$. In this case we obtain

We deduce from these estimates that

Using the result of Exercise 5.2.5 we conclude that

This implies that it suffices to obtain the required L p bound for the smoothly truncated maximal singular integral operator $$\widetilde{T}_{\varOmega }^{({\ast}{\ast})}$$.

Let K j , Ω j , and T j be as in the previous theorem, and let F j be the Riesz transform of the function $$\varOmega (x/\vert x\vert )\varPhi (x)\vert x\vert ^{-n}$$. Let f ∈ L p (R n ). A calculation yields the identity

where in the last step we used Proposition 5.1.16. Therefore we may write

(5.2.38)

where

It follows from the definitions of F j and K j that

whenever | z | ≥ 1. But using the mean value theorem, the last expression is easily seen to be bounded by

whenever | z | ≥ 1. Using this estimate, we obtain that the jth term in $$A_{2}^{(\varepsilon,N)}(f)(x)$$ is bounded by

It follows that for functions f in L p we have

in view of Theorem 2.​1.​10.​ (M here is the Hardy–Littlewood maximal operator.) By Theorem 2.​1.​6, M maps L p (R n ) to itself with norm bounded by a dimensional constant multiple of $$\max (1,(p - 1)^{-1})$$. Since by Remark 5.2.9 the norm $$\big\|R_{j}\big\|_{L^{p}\rightarrow L^{p}}$$ is controlled by a dimensional constant multiple of $$\max (p,(p - 1)^{-1})$$, it follows that

(5.2.39)

Next, recall that in the proof of Theorem 5.2.10 we showed that

where Ω j are integrable functions on S n−1 that satisfy

(5.2.40)

Consequently, for functions f in L p (R n ) we have

and by Remark 5.2.9 this last expression has L p norm at most a dimensional constant multiple of $$\big\|\varOmega _{j}\big\|_{L^{1}}\max (p,(p - 1)^{-1})\big\|R_{j}(f)\big\|_{L^{p}}$$. It follows that

(5.2.41)

Finally, we turn our attention to the term $$A_{1}^{(\varepsilon,N)}(f)$$. To prove the required estimate, we first show that there exist nonnegative homogeneous of degree zero functions G j on R n that satisfy

(5.2.42)

and

(5.2.43)

To prove (5.2.42), first note that if | x | ≤ 1∕8, then

We now fix an x satisfying 1∕8 ≤ | x | ≤ 1 and we write

where

But since 1∕8 ≤ | x | ≤ 1, we see that

and that

For P ₂(x) we use the estimate $$\vert \varPhi (y) -\varPhi (x)\vert \leq C\vert x - y\vert $$ to obtain

Recall that $$K_{j}(x) =\varOmega _{j}(x/\vert x\vert )\vert x\vert ^{-n}$$. We now set

(5.2.44)

and we observe that G j is a homogeneous of degree zero function, it satisfies (5.2.42), and it is integrable over the annulus $$\frac{1} {2} \leq \vert x\vert \leq 2$$. To verify the last assertion, we split up the double integral

into the pieces 1∕4 ≤ | y | ≤ 4, | y |  > 4, and | y |  < 1∕4. The part of I where 1∕4 ≤ | y | ≤ 4 is pointwise bounded by a constant multiple of

which is pointwise controlled by a constant multiple of $$\|\varOmega \|_{L^{1}}$$. In the part of I where | y |  > 4 we use that | x − y | −n+1 ≤ ( | y | ∕2)−n+1 to obtain rapid decay in y and hence a bound by a constant multiple of $$\|\varOmega \|_{L^{1}}$$. Finally, in the part of I where | y |  < 1∕4 we use that | x − y | −n+1 ≤ (1∕4)−n+1, and then we also obtain a similar bound. It follows from (5.2.44) and (5.2.40) that

Since G j is homogeneous of degree zero, we deduce (5.2.43).

To complete the proof, we argue as follows:

Using (5.2.43) together with the L p boundedness of the Riesz transforms and of M θ we obtain

(5.2.45)

Combining (5.2.45), (5.2.39), and (5.2.41), we obtain the required conclusion. □ 

The following corollary is a consequence of Theorem 5.2.11.

Corollary 5.2.12.

Let Ω be as in Theorem 5.2.11 . Then for 1 < p < ∞ and f in $$L^{p}(\mathbf{R}^{n})$$ the functions $$T_{\varOmega }^{(\varepsilon,N)}(f)$$ converge to T Ω (f) in L p and almost everywhere as $$\varepsilon \rightarrow 0$$ and N →∞.

Proof.

The a.e. convergence is a consequence of Theorem 2.​1.​14.​ The L p convergence is a consequence of the Lebesgue dominated convergence theorem since for $$f \in L^{p}(\mathbf{R}^{n})$$ we have that $$\vert T_{\varOmega }^{(\varepsilon,N)}(f)\vert \leq T_{\varOmega }^{({\ast}{\ast})}(f)$$ and $$T_{\varOmega }^{({\ast}{\ast})}(f)$$ is in $$L^{p}(\mathbf{R}^{n})$$. □ 

Exercises

5.2.1. Show that the directional Hilbert transform $$\mathcal{H}_{\theta }$$ is given by convolution with the distribution $$w_{\theta }$$ in $$\mathcal{S}'(\mathbf{R}^{n})$$ defined by

Compute the Fourier transform of w θ and prove that $$\mathcal{H}_{\theta }$$ maps L ¹(R n ) to L 1, ∞ (R n ).

5.2.2. Extend the definitions of W Ω and T Ω to Ω = d μ a finite signed Borel measure on S n−1 with mean value zero. Compute the Fourier transform of W d μ and find a necessary and sufficient condition on measures d μ so that T d μ is L ² bounded. Notice that the directional Hilbert transform $$\mathcal{H}_{\theta }$$ is a special case of such an operator T d μ .

5.2.3. Use the inequality A B ≤ AlogA + e B for A ≥ 1 and B > 0 to prove that if Ω satisfies (5.2.24) then it must satisfy (5.2.16). Conclude that if | Ω | log⁺ | Ω | is in L ¹(S n−1), then T Ω is L ² bounded.

[Hint: Use that $$\int _{\mathbf{S}^{n-1}}\vert \xi \cdot \theta \vert ^{-\alpha }\,d\theta$$ converges when α < 1. See Appendix D.3.]

5.2.4. Let Ω be a nonzero integrable function on S n−1 with mean value zero. Let f ≥ 0 be nonzero and integrable over R n . Prove that T Ω (f) is not in L ¹(R n ).

[Hint: Show that $$\widehat{T_{\varOmega }(f)}$$ cannot be continuous at zero.]

5.2.5. Let θ ∈ S n−1. Use an identity similar to (5.2.18) to show that the maximal operators

are $$L^{p}(\mathbf{R}^{n})$$ bounded for 1 < p < ∞ with norm at most 3 p (p − 1)⁻¹.

5.2.6. For Ω ∈ L ¹(S n−1) and f locally integrable on R n , define

Apply the method of rotations to prove that M Ω maps $$L^{p}(\mathbf{R}^{n})$$ to itself for 1 < p < ∞.

5.2.7. Let Ω(x, θ) be a function on R n ×S n−1 satisfying

(a) Ω(x, −θ) = −Ω(x, θ) for all x and θ.

(b) $$\sup _{x}\vert \varOmega (x,\theta )\vert $$ is in L ¹(S n−1).

Use the method of rotations to prove that

is bounded on $$L^{p}(\mathbf{R}^{n})$$ for 1 < p < ∞.

5.2.8. Let Ω ∈ L ¹(S n−1) have mean value zero. Prove that if T Ω maps $$L^{p}(\mathbf{R}^{n})$$ to $$L^{q}(\mathbf{R}^{n})$$, then p = q.

[Hint: Use dilations.]

5.2.9. Prove that for all 1 < p < ∞ there exists a constant A p  > 0 such that for every complex-valued $$\mathcal{C}^{2}(\mathbf{R}^{2})$$ function f with compact support we have the bound

5.2.10. (a) Let $$\varDelta =\sum _{ j=1}^{n}\partial _{x_{j}}^{2}$$ be the usual Laplacian on R n . Prove that for all 1 < p < ∞ there exists a constant A p  > 0 such that for all $$\mathcal{C}^{2}$$ functions f with compact support we have the bound

(b) Let $$\varDelta ^{m} =\overbrace{\varDelta \circ \ldots \circ \varDelta }^{m\ \text{times}}$$. Show that for any 1 < p < ∞ there exists a C p  > 0 such that for all f of class $$\mathcal{C}^{2m}$$ with compact support and all differential monomials ∂ x α of order | α |  = 2m we have

5.2.11. Use the same idea as in Lemma 5.2.5 to show that if f is continuous on [0, ∞), differentiable in (0, ∞), and satisfies

for all a > 0, then

5.2.12. Let Ω o be an odd integrable function on S n−1 and Ω e an even function on S n−1 that satisfies (5.2.24). Let f be a function supported in a ball B in R n . Prove that (a) If | f | log⁺ | f | is integrable over a ball B, then $$T_{\varOmega _{o}}(f)$$ and $$T_{\varOmega _{o}}^{({\ast}{\ast})}(f)$$ are integrable over B. (b) If | f | (log⁺ | f | )² is integrable over a ball B, then $$T_{\varOmega _{e}}(f)$$ and $$T_{\varOmega _{e}}^{({\ast}{\ast})}(f)$$ are integrable over B. [Hint: Use Exercise 1.3.7.]

5.2.13. ([324]) Let Ω be integrable on S n−1 with mean value zero. Use Jensen’s inequality to show that for some C > 0 and every radial function f ∈ L ²(R n ) we have

This inequality subsumes that T Ω is well defined on radial L ²(R n ) functions.

5.3 The Calderón–Zygmund Decomposition and Singular Integrals

The behavior of singular integral operators on L ¹(R n ) is a more subtle issue than that on L p for 1 < p < ∞. It turns out that singular integrals are not bounded from L ¹ to L ¹. See Example 5.1.3 and also Exercise 5.2.4. In this section we see that singular integrals map L ¹ into the larger space L 1, ∞ . This result strengthens their L p boundedness.

5.3.1 The Calderón–Zygmund Decomposition

To make some advances in the theory of singular integrals, we need to introduce the Calderón–Zygmund decomposition. This is a powerful stopping-time construction that has many other interesting applications. We have already encountered an example of a stopping-time argument in Section 2.​1.​

Recall that a dyadic cube in R n is the set

where $$k,m_{1},\ldots,m_{n} \in \mathbf{Z}$$. Two dyadic cubes are either disjoint or related by inclusion.

Theorem 5.3.1.

Let f ∈ L ¹ ( R n ) and α > 0. Then there exist functions g and b on R n such that

(1)

f = g + b.

(2)

$$\quad \|g\|_{L^{1}} \leq \| f\|_{L^{1}}$$ and $$\|g\|_{L^{\infty }} \leq 2^{n}\alpha$$ .

(3)

$$\quad b =\sum _{j}b_{j}$$ , where each b j is supported in a dyadic cube Q j . Furthermore, the cubes Q k and Q j are disjoint when j ≠ k.

(4)

(5)

$$\quad \|b_{j}\|_{L^{1}} \leq 2^{n+1}\alpha \vert Q_{j}\vert $$ .

(6)

$$\quad \sum _{j}\vert Q_{j}\vert \leq \alpha ^{-1}\|f\|_{L^{1}}$$ .

Remark 5.3.2.

This decomposition is called the Calderón–Zygmund decomposition of f at height α. The function g is called the good function of the decomposition, since it is both integrable and bounded; hence the letter g. The function b is called the bad function , since it contains the singular part of f (hence the letter b), but it is carefully chosen to have mean value zero. It follows from (1) and (2) that the bad function b is integrable and satisfies

By (2) the good function is integrable and bounded; hence it lies in all the L p spaces for 1 ≤ p ≤ ∞. More specifically, we have the following estimate:

(5.3.1)

Proof.

Decompose R n into a mesh of disjoint dyadic cubes of the same size such that

for every cube Q in the mesh. Call these cubes of zero generation. Subdivide each cube of zero generation into 2 n congruent cubes by bisecting each of its sides. We now have a new mesh of dyadic cubes, which we call of generation one. Select a cube Q of generation one if

(5.3.2)

Let S ⁽¹⁾ be the set of all selected cubes of generation one. Now subdivide each nonselected cube of generation one into 2 n congruent subcubes by bisecting each side and call these cubes of generation two. Then select all cubes Q of generation two if (5.3.2) holds. Let S ⁽²⁾ be the set of all selected cubes of generation two. Repeat this procedure indefinitely.

The set of all selected cubes $$\bigcup _{m=1}^{\infty }S^{(m)}$$ is countable and is exactly the set of the cubes Q j proclaimed in the proposition. Note that in some instances this set may be empty, in which case b = 0 and g = f. Let us observe that the selected cubes are disjoint, for otherwise some Q k would be a proper subset of some Q j , which is impossible since the selected cube Q j was never subdivided. Now define

$$b =\sum _{j}b_{j}$$, and g = f − b.

For a selected cube Q j there exists a unique nonselected cube Q′ with twice its side length that contains Q j . Let us call this cube the parent of Q j . Since the parent Q′ of Q j was not selected, we have $$\vert Q'\vert ^{-1}\int _{Q'}\vert f\vert \,dx \leq \alpha$$. Then

Consequently,

which proves (5). To prove (6), simply observe that

Next we need to obtain the estimates concerning g. We obviously have

(5.3.3)

On the cube Q j , g is equal to the constant $$\vert Q_{j}\vert ^{-1}\int _{Q_{j}}f\,dx$$, and this is bounded by 2 n α. It suffices to show that g is bounded outside the union of the Q j ’s. Indeed, for each $$x \in \mathbf{R}^{n}\setminus \bigcup _{j}Q_{j}$$ and for each $$k = 0,1,2,\ldots$$ there exists a unique nonselected dyadic cube Q x (k) of generation k that contains x. Then for each k ≥ 0, we have

The intersection of the closures of the cubes Q x (k) is the singleton {x}. Using Corollary 2.​1.​16, we deduce that for almost all $$x \in \mathbf{R}^{n}\setminus \bigcup _{j}Q_{j}$$ we have

Since these averages are at most α, we conclude that | f | ≤ α a.e. on $$\mathbf{R}^{n}\setminus \bigcup _{j}Q_{j}$$, hence | g | ≤ α a.e. on this set. Finally, it follows from (5.3.3) that $$\|g\|_{L^{1}} \leq \| f\|_{L^{1}}$$. This finishes the proof of the theorem. □ 

We now apply the Calderón–Zygmund decomposition to obtain weak type (1, 1) bounds for a wide class of singular integral operators that includes the operators T Ω we studied in the previous section.

5.3.2 General Singular Integrals

The kernels of the general singular integrals we will study are tempered distributions that coincide with functions away from the origin. The setup as follows. Let K be a measurable function defined on R n ∖{0} that is integrable on compact subsets of R n ∖{0} and satisfies the size condition

(5.3.4)

This condition is less restrictive than the standard size estimate

(5.3.5)

but it is strong enough to capture size properties of kernels K(x) = Ω(x∕ | x | )∕ | x |  n , where Ω ∈ L ¹(S n−1). We also note that condition (5.3.4) is equivalent to

(5.3.6)

See Exercise 5.3.1.

The size condition (5.3.4) is sufficient to make the restriction of K(x) on | x |  > δ a tempered distribution (for any δ > 0). Indeed, for $$\varphi \in \mathcal{S}(\mathbf{R}^{n})$$ we have

and this expression is bounded by a constant times a finite sum of Schwartz seminorms of $$\varphi$$.

We are interested in tempered distributions W on R n that extend the function K defined on R n ∖{0} and have the form

(5.3.7)

for some sequence $$\delta _{j} \downarrow 0$$ as j → ∞. It is not hard to see that there exists a tempered distribution W satisfying (5.3.7) for all $$\varphi \in \mathcal{S}(\mathbf{R}^{n})$$ if and only if

(5.3.8)

exists. See Exercise 5.3.2. If such a distribution W exists it may not be unique, since it depends on the choice of the sequence δ j . Two different sequences tending to zero may give two different tempered distributions W of the form (5.3.7), both coinciding with the function K on $$\mathbf{R}^{n}\setminus \{0\}$$. See Example 5.4.2 and Remark 5.4.3.

If condition (5.3.8) is satisfied, we can define

(5.3.9)

and the limit exists as j → ∞ for all $$\varphi \in \mathcal{S}(\mathbf{R}^{n})$$ and is equal to

Moreover, the previous calculations show that W is an element of $$\mathcal{S}'(\mathbf{R}^{n})$$.

Next we assume that the given function K on R n ∖{0} satisfies a certain smoothness condition. There are three kinds of smoothness conditions that we encounter: first, the gradient condition

(5.3.10)

next, the weaker Lipschitz condition ,

(5.3.11)

and finally the even weaker smoothness condition

(5.3.12)

for some A ₂ < ∞. One should verify that (5.3.12) is a weaker condition than (5.3.11), which in turn is weaker than (5.3.10). Condition (5.3.12) is often referred to as Hörmander’s condition.

5.3.3 L r Boundedness Implies Weak Type (1, 1) Boundedness

This next theorem provides the most classical application of the Calderón–Zygmund decomposition.

Theorem 5.3.3.

Let K be a function on R n ∖{0} that satisfies (5.3.4) ¹ and (5.3.12) for some A ₁ ,A ₂ < ∞. Let W be an element of $$\mathcal{S}'(\mathbf{R}^{n})$$ related to K as in (5.3.7) . Suppose that the operator T given by convolution with W has a bounded extension that maps L r ( R n ) to itself with norm B for some 1 < r ≤∞. Then T has an extension that maps L ¹ ( R n ) to L 1,∞ ( R n ) with norm

(5.3.13)

and T also extends to a bounded operator from $$L^{p}(\mathbf{R}^{n})$$ to itself for 1 < p < ∞ with norm

(5.3.14)

where C n ,C n ′ are constants that depend on the dimension but not on r or p.

Proof.

We discuss the case r < ∞ and we refer to Exercise 5.3.7 for the case r = ∞. Let α > 0 be given. We fix a step function f given as a finite linear combination of characteristic functions of disjoint dyadic intervals. The class of such functions is dense in all the L p spaces. Once (5.3.13) is obtained for such functions, a density argument gives that T admits an extension on L ¹ that also satisfies (5.3.13). Therefore it suffices to prove (5.3.13) for such a function f.

Apply the Calderón–Zygmund decomposition to f at height γ α, where γ is a positive constant to be chosen later. That is, write the function f as the sum

where conditions (1)–(6) of Theorem 5.3.1 are satisfied with the constant α replaced by γ α. Since f is a finite linear combination of characteristic functions of disjoint dyadic cubes, there are only finitely many cubes Q j that appear in the Calderón–Zygmund decomposition to f. Each b j is supported in a dyadic cube Q j with center y j and the Q j ’s are pairwise disjoint. We denote by ℓ(Q) the side length of a cube Q. Let Q j ∗ be the unique cube with sides parallel to the axes having the same center as Q j and having side length $$\ell(Q_{j}^{{\ast}}) = 2\sqrt{n}\,\ell(Q_{j}).$$ Because of the form of f, each b j is a bounded function supported in $$\overline{Q_{j}}$$, hence it is in L r , thus each T(b j ) is a well-defined L r function. We observe that for all j and all $$x\notin Q_{j}^{{\ast}}$$ we have

where the last integral converges absolutely. This is a consequence of the Lebesgue dominated convergence theorem, based on the facts that b j is bounded, that K is integrable over any compact annulus that does not contain the origin (cf. (5.3.4)), and that x − Q j is contained in such a compact annulus, since $$x\notin Q_{j}^{{\ast}}$$.

Next we use the cancellation of b j in the following way:

where we used (5.3.12) since if $$x \in -y_{j} + (Q_{j}^{{\ast}})^{c}$$ then $$\vert x\vert \geq \frac{1} {2}\ell(Q_{j}^{{\ast}}) = \sqrt{n}\,\ell(Q_{ j})$$ and since y − y j  ∈ −y j + Q j we have $$\vert y - y_{j}\vert \leq \frac{\sqrt{n}} {2} \,\ell(Q_{j})$$, thus | x | ≥ 2 | y − y j  | . See Figure 5.2.

Fig. 5.2

The cubes − y j + Q j and − y j + Q j ∗.

Thus we proved that

an inequality we use below. We have

Choosing $$\gamma = 2^{-(n+1)}B^{-1}$$, we deduce the weak type (1, 1) estimate (5.3.13) for T with $$C_{n} = 2 + 2^{n+1}(2\sqrt{n})^{n} + 2^{n+2}$$.

It follows from Exercise 1.3.2 that T has an extension that maps L p to L p with bound at most C n ′(A ₂ + B)(p − 1)−1∕p when 1 < p < r. The adjoint operator T ∗ of T, defined by

has a kernel that coincides with the function $$K^{{\ast}}(x) = \overline{K(-x)}$$ on R n ∖{0}. We notice that since K satisfies (5.3.12), then so does K ∗ and with the same bound. Therefore, T ∗, which maps L r′ to L r′, has a kernel that satisfies (5.3.12). By the preceding argument, T ∗ maps L p′ to L p′ with bound at most $$C_{n}'(A_{2} + B)(p - 1)^{-1/p}$$ whenever 1 < p′ < r′. By duality this yields that T maps $$L^{p}(\mathbf{R}^{n})$$ to $$L^{p}(\mathbf{R}^{n})$$ with bound at most C n ′(A ₂ + B)(p − 1)1−1∕p whenever r′ < p < ∞. Using interpolation we obtain that T maps L p to itself with norm at most $$C_{n}'(A_{2} + B)\max ((p - 1)^{-1/p},(p - 1)^{1-1/p})$$ for p in the interval (r, r′), which is nonempty only if r < 2. Then (5.3.14) holds since $$\max ((p - 1)^{-1/p},(p - 1)^{1-1/p}) \leq \max ((p - 1)^{-1},p)$$. □ 

5.3.4 Discussion on Maximal Singular Integrals

In this subsection we introduce maximal singular integrals and we derive their boundedness under certain smoothness conditions on the kernels, assuming boundedness of the associated linear operator.

Suppose that K is a kernel on R n ∖{0} that satisfies the size condition

(5.3.15)

for x ≠ 0. Then for any $$\varepsilon > 0$$ the function $$K^{(\varepsilon )}(x) = K(x)\chi _{\vert x\vert \geq \varepsilon }$$ lies in L p′(R n ) (with norm bounded by $$c_{p,n}\,A_{1}\varepsilon ^{-n/p}$$) for all 1 ≤ p < ∞. Consequently, by Hölder’s inequality, the integral

converges absolutely for all x ∈ R n and all f ∈ L p (R n ), when 1 ≤ p < ∞.

Let $$f \in \bigcup _{1\leq p<\infty }L^{p}(\mathbf{R}^{n})$$. We define the truncated singular integrals $$T^{(\varepsilon )}(f)$$ associated with the kernel K by setting

we also define the maximal truncated singular integral operator associated with K by setting

This operator is well defined, but possibly infinite, for certain points in R n .

We now consider the situation in which the kernel K satisfies an integrability condition over concentric annuli centered at the origin, a condition that is certainly a weaker condition than (5.3.15). Precisely, suppose that K is a measurable function on R n ∖{0}, that is integrable on compact subsets of R n ∖{0}, for which there is a constant A ₁ < ∞ such that

(5.3.16)

Such kernels may not be integrable to the power p′ > 1 over the region $$\vert x\vert \geq \varepsilon$$. For this reason, it is not possible to define $$T^{(\varepsilon )}$$ as an absolutely convergent integral. To overcome this difficulty, we consider double truncations. We define the doubly truncated kernel $$K^{(\varepsilon,N)}$$ by setting

(5.3.17)

A repeated application of (5.3.16) yields that

which implies that $$K^{(\varepsilon,N)}$$ is integrable over concentric annuli centered at the origin. Next, we define the doubly truncated singular integrals $$T^{(\varepsilon,N)}$$ by setting

and we observe that these operators are well defined when f in L p , for 1 ≤ p ≤ ∞. Indeed, Theorem 1.​2.​10 yields that

for functions f in L p , 1 ≤ p ≤ ∞. Consequently, for almost every x ∈ R n we have

For functions in $$\bigcup _{1\leq p\leq \infty }L^{p}(\mathbf{R}^{n})$$ we define the doubly truncated maximal singular integral operator T (∗∗) associated with K by setting

(5.3.18)

For such functions and for almost all x ∈ R n , T (∗∗)(f)(x) is well defined, but potentially infinite.

One observation is that under condition (5.3.16), one can also define T (∗)(g) for general integrable functions g with compact support. In this case, say that the ball B(0, R) contains the support of g. Let x ∈ B(0, M) and N = M + R. Then $$\vert T^{(\varepsilon )}(g)(x)\vert \leq \vert g\vert {\ast}\vert K^{(\varepsilon,N)}\vert (x)$$, which is finite a.e. as the convolution of two L ¹ functions; consequently, the integral defining $$T^{(\varepsilon )}(g)(x)$$ converges absolutely for all x ∈ B(0, R). Since R > 0 is arbitrary, $$T^{(\varepsilon )}(g)(x)$$ is defined and finite for almost all x ∈ R n .

Obviously T (∗) and T (∗∗) are related. If K satisfies condition (5.3.15), then

which implies that

for all $$f \in \bigcup _{1\leq p<\infty }L^{p}$$. Also, $$T^{(\varepsilon,N)}(f) = T^{(\varepsilon )}(f) - T^{(N)}(f)$$; hence

Therefore, for kernels satisfying (5.3.15), T (∗∗) and T (∗) are comparable and the boudnedness properties of T (∗∗) and T (∗) are equivalent

Theorem 5.3.4.

(Cotlar’s inequality) Let 0 < A ₁ ,A ₂ ,A ₃ < ∞ and suppose that K is defined on R n ∖{0} and satisfies the size condition,

(5.3.19)

the smoothness condition

(5.3.20)

whenever |x|≥ 2|y| > 0, and the cancellation condition

(5.3.21)

Let W in $$\mathcal{S}'(\mathbf{R}^{n})$$ be related to K via (5.3.7) and let T be the operator given by convolution with W. Then there is a constant C n,δ such that the following inequality is valid:

(5.3.22)

for all $$f \in \mathcal{S}(\mathbf{R}^{n})$$ , where M is the Hardy–Littlewood maximal operator. Consequently, T (∗) is bounded on $$L^{p}(\mathbf{R}^{n})$$ when 1 < p < ∞.

Proof.

Let $$\varphi$$ be a radially decreasing smooth function with integral 1 supported in the ball B(0, 1∕2). For a function g and $$\varepsilon > 0$$ we use the notation $$g_{\varepsilon }(x) =\varepsilon ^{-n}g(\varepsilon ^{-1}x)$$. For a distribution W we define $$W_{\varepsilon }$$ analogously, i.e. as the unique distribution with the property $$\langle W_{\varepsilon },\psi \rangle =\varepsilon ^{-n}\langle W,\psi _{\varepsilon ^{-1}}\rangle$$. We begin by observing that $$K_{\varepsilon ^{-1}}(x) =\varepsilon ^{n}K(\varepsilon x)$$ satisfies (5.3.19), (5.3.20), and (5.3.21) uniformly in $$\varepsilon > 0$$.

Set, as before, $$K^{(\varepsilon )}(x) = K(x)\chi _{\vert x\vert \geq \varepsilon }$$. Fix $$f \in \mathcal{S}(\mathbf{R}^{n})$$ for some 1 < p < ∞. Obviously we have

(5.3.23)

Next we prove the following estimate for all $$\varepsilon > 0$$:

(5.3.24)

for all x ∈ R n . Indeed, for | x | ≥ 1 we express the left-hand side in (5.3.24) as

Since $$\varphi$$ is supported in | y | ≤ 1∕2, we have | x | ≥ 2 | y | , and condition (5.3.20) yields that the expression on the left-hand side of (5.3.24) is bounded by

which proves (5.3.24) in the case | x | ≥ 1. When | x |  < 1, the left-hand side of (5.3.24) equals

(5.3.25)

for some sequence $$\delta _{j} \downarrow 0$$; see the discussion in Section 5.3.2. The expression in (5.3.25) is equal to

where

In I ₁ we have $$1/8 \leq \vert x - y\vert \leq 1 + 1/2 = 3/2$$; hence I ₁ is bounded by a multiple of A ₁. Since $$\vert \varphi (x) -\varphi (y)\vert \leq c\vert x - y\vert $$, the same is valid for I ₂. Finally, I ₃ is bounded by a multiple of A ₃. Combining these facts yields the proof of (5.3.24) in the case | x |  < 1 as well.

Use Corollary 2.​1.​12 to deduce that

Finally, take the supremum over $$\varepsilon > 0$$ in (5.3.23) and use (5.3.24) and Corollary 2.​1.​12 one more time to deduce the estimate

where C depends on n and δ; this concludes the proof of (5.3.22) for all functions $$f \in \mathcal{S}(\mathbf{R}^{n})$$. Thus T (∗) is bounded on L p , 1 < p < ∞, when restricted to Schwartz functions.

Now given a general function g in $$L^{p}(\mathbf{R}^{n})$$ we find a sequence h j in $$\mathcal{S}(\mathbf{R}^{n})$$ such that $$\|h_{j} - g\|_{L^{p}} \rightarrow 0$$ as j → ∞. Then we have the pointwise estimate

for all $$\varepsilon \geq \varepsilon _{0}$$. Taking the supremum over $$\varepsilon \geq \varepsilon _{0}$$ and then L p norm over the ball B(0, R), we obtain

Now we let j → ∞ first, and then R → ∞ and $$\varepsilon _{0} \rightarrow 0$$ to deduce the boundedness of T (∗) on $$L^{p}(\mathbf{R}^{n})$$ via the Lebesgue monotone convergence theorem. □ 

5.3.5 Boundedness for Maximal Singular Integrals Implies Weak Type (1, 1) Boundedness

We now state and prove a result analogous to that in Theorem 5.3.3 for maximal singular integrals.

Theorem 5.3.5.

Let K(x) be function on R n ∖{0} satisfying (5.3.4) with constant A ₁ < ∞ and Hörmander’s condition (5.3.12) with constant A ₂ < ∞. Suppose that the operator T (∗∗) as defined in (5.3.18) maps L ² ( R n ) to itself with norm B. Then T (∗∗) maps L ¹ ( R n ) to L 1,∞ ( R n ) with norm

where C n is some dimensional constant.

Proof.

The proof of this theorem is only a little more involved than the proof of Theorem 5.3.3. We fix an L ¹(R n ) function f. We apply the Calderón–Zygmund decomposition of f at height γ α for some γ, α > 0. We then write f = g + b, where $$b =\sum _{j}b_{j}$$ and each b j is supported in some cube Q j . We define Q j ∗ as the cube with the same center as Q j and with sides parallel to the sides of Q j having length $$\ell(Q_{j}^{{\ast}}) = 5\sqrt{n}\,\ell(Q_{j})$$. This is only a minor change compared with the definition of Q j in Theorem 5.3.3. The main change in the proof is in the treatment of the term

(5.3.26)

We show that for all γ ≤ (2 n+5 A ₁)⁻¹ we have

(5.3.27)

Let us conclude the proof of the theorem assuming for the moment the validity of (5.3.27). As in the proof of Theorem 5.3.3, we can show that

Combining this estimate with (5.3.27) and choosing

we obtain the required estimate

with $$C_{n} = 2^{-3} + (5\sqrt{n})^{n}2^{n+5} + 2^{n+8}$$.

It remains to prove (5.3.27). This estimate will be a consequence of the fact that for $$x \in \big (\bigcup _{j}Q_{j}^{{\ast}}\big)^{c}$$ we have the key inequality

(5.3.28)

where

and y j is the center of Q j .

If we had (5.3.28), then we could easily derive (5.3.27). Indeed, fix a γ satisfying $$\gamma \leq (2^{n+5}A_{1})^{-1}$$. Then we have $$2^{n+3}\alpha \gamma A_{1} < \frac{\alpha } {3}$$, and using (5.3.28), we obtain

(5.3.29)

since $$\frac{\alpha }{2} = \frac{\alpha } {3} + \frac{\alpha } {12} + \frac{\alpha } {12}$$. We have

Fig. 5.3

The cubes Q j and Q j ∗.

(5.3.30)

where we used the fact that if x ∈ (Q j ∗) c , then $$\vert x - y_{j}\vert \geq \frac{1} {2}\ell(Q_{j}^{{\ast}}) = \frac{5} {2}\sqrt{n}\,\ell(Q_{j})$$. But since $$\vert y - y_{j}\vert \leq \frac{\sqrt{n}} {2} \,\ell(Q_{j})$$, this implies that | x − y j  | ≥ 2 | y − y j  | . Here we used the fact that the diameter of a cube is equal to $$\sqrt{n}$$ times its side length. Likewise, we obtain that

(5.3.31)

Combining (5.3.30) and (5.3.31) with (5.3.29) yields (5.3.27).

Therefore, the main task of the proof is to prove (5.3.28). Since $$b =\sum _{j}b_{j}$$, to estimate T (∗∗)(b), it suffices to estimate each $$\vert T^{(\varepsilon,N)}(b_{j})\vert $$ uniformly in $$\varepsilon$$ and N. To achieve this we use the estimate

(5.3.32)

noting that the truncated singular integrals $$T^{(\varepsilon )}(b_{j})$$ are well defined. Indeed, say x lies in a compact set K ₀. Pick M such that K ₀ − Q j is contained in a ball B(0, M). Then

which is finite a.e. as the convolution of two L ¹ functions; thus the integral defining $$T^{(\varepsilon )}(b_{j})(x)$$ converges absolutely and the expression $$T^{(\varepsilon )}(b_{j})(x)$$ is well defined for almost all x.

We work with $$T^{(\varepsilon )}$$ and we note that T (N) can be treated similarly. Fix $$x\notin \bigcup _{j}Q_{j}^{{\ast}}$$ and $$\varepsilon > 0$$ and define

Note that

whenever $$x\notin \bigcup _{j}Q_{j}^{{\ast}}$$ and $$j \in J_{1}(x,\varepsilon )$$. Also note that

whenever $$x\notin \bigcup _{j}Q_{j}^{{\ast}}$$, $$j \in J_{2}(x,\varepsilon )$$ and y ∈ Q j . Therefore,

but since

(5.3.33)

it suffices to estimate the term

We now make some geometric observations; see Figure 5.3. Fix $$\varepsilon > 0$$ and a cube Q j with $$j \in J_{3}(x,\varepsilon )$$; recall that x lies in $$(\bigcup _{j}Q_{j}^{{\ast}})^{c}$$. Then we have

(5.3.34)

Since $$j \in J_{3}(x,\varepsilon )$$, there exists a y ₀ ∈ Q j with

Using (5.3.34), we obtain that for any y ∈ Q j we have

Therefore, we have proved that

Letting

we note that in view of property (5) of the Calderón–Zygmund decomposition (Theorem 5.3.1), the estimate

holds. Then

The last estimate, together with (5.3.33), with (5.3.32), and with the analogous estimate for $$\sup _{N>0}\vert T^{(N)}(b_{j})(x)\vert $$ (which is similarly obtained), yields (5.3.28). □ 

The value of the previous theorem lies in the following: Since we know that for some sequences $$\varepsilon _{j} \downarrow 0$$, N j ↑ ∞ the pointwise limit $$T^{(\varepsilon _{j},N_{j})}(f)$$ exists a.e. for all f in a dense subclass of L ¹, then Theorem 5.3.5 allows us to deduce that $$T^{(\varepsilon _{j},N_{j})}(f)$$ exists a.e. for all f in L ¹(R n ).

If the singular integrals have kernels of the form $$\varOmega (x/\vert x\vert )\vert x\vert ^{-n}$$ with Ω in L ∞ , such as the Hilbert transform and the Riesz transforms, then the upper truncations are not needed for K in (5.3.17). In this case

is well defined for $$f \in \bigcup _{1\leq p<\infty }L^{p}(\mathbf{R}^{n})$$ by Hölder’s inequality and is equal to

Corollary 5.3.6.

The maximal Hilbert transform H (∗) and the maximal Riesz transforms R j (∗) are weak type (1,1). Secondly,$$\lim _{\varepsilon \rightarrow 0}H^{(\varepsilon )}(f)$$ and $$\lim _{\varepsilon \rightarrow 0}R_{j}^{(\varepsilon )}(g)$$ exist a.e. for all $$f \in L^{1}(\mathbf{R})$$ and $$g \in L^{1}(\mathbf{R}^{n})$$ , as $$\varepsilon \rightarrow 0$$ .

Proof.

Since the kernels 1∕x on R and x j ∕ | x |  n on R n satisfy (5.3.10), the first statement in the corollary is an immediate consequence of Theorem 5.3.5. The second statement follows from Theorem 2.​1.​14 and Corollary 5.2.8, since these limits exist for Schwartz functions. □ 

Corollary 5.3.7.

Under the hypotheses of Theorem 5.3.5, T (∗∗) maps $$L^{p}(\mathbf{R}^{n})$$ to itself for 1 < p < 2 with norm

where C n is some dimensional constant.

Exercises

5.3.1. Let A ₁ be defined in (5.3.4). Prove that

thus the expressions in (5.3.6) and (5.3.4) are equivalent.

5.3.2. Suppose that K is a locally integrable function on R n ∖{0} that satisfies (5.3.4). Suppose that δ j ↓ 0. Prove that the principal value operation

defines a distribution in $$\mathcal{S}'(\mathbf{R}^{n})$$ if and only if the following limit exists:

5.3.3. Suppose that a function K on R n ∖{0} satisfies condition (5.3.4) with constant A ₁ and condition (5.3.12) with constant A ₂.

(a) Show that the functions $$K(x)\chi _{\vert x\vert \geq \varepsilon }$$ also satisfy condition (5.3.12) uniformly in $$\varepsilon > 0$$ with constant A ₁ + A ₂.

(b) Obtain the same conclusion for the upper truncations $$K(x)\chi _{\vert x\vert \leq N}\!$$.

(c) Deduce a similar conclusion for the double truncations $$K^{(\varepsilon,N)}(x) = K(x)\chi _{\varepsilon \leq \vert x\vert \leq N}$$.

5.3.4. Modify the proof of Theorem 5.3.5 to prove that if T (∗∗) maps L r to $$L^{r,\infty }$$ for some $$1 < r < \infty $$, and K satisfies condition (5.3.12), then T (∗∗) maps L ¹ to L 1, ∞ .

5.3.5. Assume that T is a linear operator acting on measurable functions on R n such that whenever a function f is supported in a cube Q, then T(f) is supported in a fixed multiple of Q.

(a) Suppose that T maps L p to itself for some 1 < p < ∞ with norm B. Prove that T extends to a bounded operator from L ¹ to L 1, ∞ with norm a constant multiple of B.

(b) Suppose that T maps L p to L q for some 1 < q < p < ∞ with norm B. Prove that T extends to a bounded operator from L ¹ to L s, ∞ with norm a multiple of B, where

5.3.6. (a) Let 1 < q < ∞. Show that the good function g in Theorem 5.3.1 lies in the Lorentz space L q, 1 and that $$\|g\|_{L^{q,1}} \leq C_{n,q}\alpha ^{1/q'}\|f\|_{L^{1}}^{1/q}$$ for some constant C n, q .

(b) Let 1 < r < ∞. Obtain a generalization of Theorem 5.3.3 in which the assumption that T maps L r to L r is replaced by that T maps L r, 1 to L r, ∞ with norm B.

(c) Let 1 < r < ∞. Obtain a further generalization of Theorem 5.3.3 in which the assumption that T maps L r to L r is replaced by that it is of restricted weak type (r, r), i.e., it satisfies

for all subsets E of R n with finite measure.

5.3.7. Let K satisfy (5.3.12) for some A ₂ > 0, let $$W \in \mathcal{S}'(\mathbf{R}^{n})$$ be an extension of K on R n as in (5.3.7), and let T be the operator given by convolution with W. Obtain the case r = ∞ in Theorem 5.3.3. Precisely, prove that if T maps $$L^{\infty }(\mathbf{R}^{n})$$ to itself with constant B, then T has an extension on L ¹ + L ∞ that satisfies

and for 1 < p < ∞ it satisfies

where C n , C n ′ are constants that depend only on the dimension.

[Hint: Apply the Calderón–Zygmund decomposition f = g + b at height α γ, where γ = (2 n+1 B)⁻¹. Since | g | ≤ 2 n α γ, observe that

For the interpolation use the result of Exercise 1.3.2.]

5.3.8. (Calderón–Zygmund decomposition on L q) Fix a function f ∈ L q (R n ) for some 1 ≤ q < ∞ and let α > 0. Then there exist functions g and b on R n such that

(1)

 f = g + b.

(2)

$$\,\|g\|_{L^{q}} \leq \| f\|_{L^{q}}$$ and $$\|g\|_{L^{\infty }} \leq 2^{\frac{n} {q} }\alpha$$.

(3)

$$\,b =\sum _{j}b_{j}$$, where each b j is supported in a cube Q j . Furthermore, the cubes Q k and Q j have disjoint interiors when j ≠ k.

(4)

$$\,\|b_{j}\|_{L^{q}}^{q} \leq 2^{n+q}\alpha ^{q}\vert Q_{j}\vert $$.

(5)

(6)

$$\,\sum _{j}\vert Q_{j}\vert \leq \alpha ^{-q}\|f\|_{L^{q}}^{q}$$.

(7)

$$\,\|b\|_{L^{q}} \leq 2^{\frac{n+q} {q} }\|f\|_{L^{q}}$$ and $$\|b\|_{L^{1}} \leq 2\alpha ^{1-q}\|f\|_{L^{q}}^{q}$$.

[Hint: Imitate the basic idea of the proof of Theorem 5.3.1, but select a cube Q if $$\big( \frac{1} {\vert Q\vert }\int _{Q}\vert f(x)\vert ^{q}\,dx\big)^{1/q} >\alpha$$. Define g and b as in the proof of Theorem 5.3.1.]

5.3.9. Let f ∈ L ¹(R n ). Then for any α > 0, prove that there exist disjoint cubes Q j in R n such that the set $$E_{\alpha } =\{ x \in \mathbf{R}^{n}:\,\, M_{c}(f)(x) >\alpha \}$$ is contained in $$\bigcup _{j}3Q_{j}$$ and $$\frac{\alpha }{4^{n}} < \frac{1} {\vert Q_{j}\vert }\int _{Q_{j}}\vert f(t)\vert \,dt \leq \frac{\alpha } {2^{n}}$$.

[Hint: For given α > 0, select all maximal dyadic cubes Q j (α) such that the average of f over them is bigger than α. Given x ∈ E α , pick a cube R that contains x such that the average of | f | over R is bigger than α and find a dyadic cube Q such that $$2^{-n}\vert Q\vert < \vert R\vert \leq \vert Q\vert $$ and that $$\int _{R\cap Q}\vert f\vert \,dx > 2^{-n}\alpha \vert R\vert $$. Conclude that Q is contained in some $$Q_{k}(4^{-n}\alpha )$$ and thus R is contained in 3Q k (4−n α). The collection of all Q j  = Q j (4−n α) is the required one.]

5.3.10. Let K(x) be a function on R n ∖{0} that satisfies | K(x) | ≤ A | x | −n . Let η(x) be a smooth function equal to 1 when | x | ≥ 2 and vanishing when | x | ≤ 1. For f ∈ L p , 1 ≤ p < ∞, define truncated singular integral operators

Show that the truncated maximal singular integral $$T^{({\ast})}(f) =\sup _{\varepsilon >0}\vert T^{(\varepsilon )}(f)\vert $$ is L p bounded for 1 < p < ∞ if and only if the smoothly truncated maximal singular integral $$T_{\eta }^{({\ast})}(f) =\sup _{\varepsilon >0}\vert T_{\eta }^{(\varepsilon )}(f)\vert $$ is L p bounded. Formulate an analogous statement for p = 1.

5.3.11. (M. Mastyło) Let 1 ≤ p < ∞. Suppose that $$T_{\varepsilon }$$ are linear operators defined on $$L^{p}(\mathbf{R}^{n})$$ such that for all $$f \in L^{p}(\mathbf{R}^{n})$$ we have $$\vert T_{\varepsilon }(f)\vert \leq A\,\varepsilon ^{-a}\|f\|_{L^{p}}$$ for some 0 < a, A < ∞. Also suppose that there is a constant C < ∞ such that the maximal operator $$T_{{\ast}}(f) =\sup _{\varepsilon >0}\vert T_{\varepsilon }(f)\vert $$ satisfies $$\big\|T_{{\ast}}(h)\|_{L^{p}} \leq C\,\|h\|_{L^{p}}$$ for all $$h \in \mathcal{S}(\mathbf{R}^{n})$$. Prove that the same inequality is valid for all $$f \in L^{p}(\mathbf{R}^{n})$$.

[Hint: For a fixed δ > 0 define $$S_{\delta }(f) =\sup _{\varepsilon >\delta }\vert T_{\varepsilon }(f)\vert $$, which is a subadditive functional on $$L^{p}(\mathbf{R}^{n})$$. For a fixed $$f_{0} \in L^{p}(\mathbf{R}^{n})$$ define a linear space $$X_{0} =\{\lambda f_{0}:\,\,\lambda \in \mathbf{C}\}$$ and a linear functional T ₀ on X ₀ by setting $$T_{0}(\lambda f_{0}) =\lambda S_{\delta }(f_{0})$$. By the Hahn–Banach theorem there is an extension $$\widetilde{T}_{0}$$ of T ₀ that satisfies $$\vert \widetilde{T}_{0}(f)\vert \leq S_{\delta }(f)$$ for all $$f \in L^{p}(\mathbf{R}^{n})$$. Since $$S_{\delta }$$ is L p is bounded on Schwartz functions with norm at most C, then so is $$\widetilde{T}_{0}$$. But $$\widetilde{T}_{0}$$ is linear and by density it is bounded on $$L^{p}(\mathbf{R}^{n})$$ with norm at most C; consequently, $$\|S_{\delta }(f_{0})\|_{L^{p}} =\| T_{0}(f_{0})\|_{L^{p}} =\|\widetilde{ T}_{0}(f_{0})\|_{L^{p}} \leq C\,\|f_{0}\|_{L^{p}}$$. The required conclusion for T ∗ follows by Fatou’s lemma.]

5.4 Sufficient Conditions for L p Boundedness

We have used the Calderón–Zygmund decomposition to prove weak type (1, 1) boundedness for singular integral and maximal singular integral operators, assuming that these operators are already L ² bounded. It is therefore natural to ask for sufficient conditions that imply L ² boundedness for such operators. Precisely, what are sufficient conditions on functions K on R n ∖{0} so that the corresponding singular and maximal singular integral operators associated with K are L ² bounded? We saw in Section 5.2 that if K has the special form $$K(x) =\varOmega (x/\vert x\vert )/\vert x\vert ^{n}$$ for some $$\varOmega \in L^{1}(\mathbf{S}^{n-1})$$ with mean value zero, then condition (5.2.16) is necessary and sufficient for the L ² boundedness of T, while the L ² boundedness of T (∗) requires the stronger smoothness condition (5.2.24).

For the general K considered in this section (for which the corresponding operator does not necessarily commute with dilations), we only give some sufficient conditions for L ² boundedness of T and T (∗∗).

Throughout this section K denotes a locally integrable function on R n ∖{0} that satisfies the “size” condition

(5.4.1)

the “smoothness” condition

(5.4.2)

and the “cancellation” condition

(5.4.3)

for some A ₁, A ₂, A ₃ > 0. As mentioned earlier, condition (5.4.2) is often referred to as Hörmander’s condition. In this section we show that these three conditions give rise to convolution operators that are bounded on L p .

5.4.1 Sufficient Conditions for L p Boundedness of Singular Integrals

We first note that under conditions (5.4.1), (5.4.2), and (5.4.3), there exists a tempered distribution W of the form (5.3.7) that coincides with K on R n ∖{0}. Indeed, condition (5.4.3) implies that there exists a sequence δ j ↓ 0 such that

exists. Using (5.3.8), we conclude that there exists such a tempered distribution W. Note that we must have | L | ≤ A ₃.

We observe that the difference of two distributions W and W′ that coincide with K on R n ∖{0} must be supported at the origin.

Theorem 5.4.1.

Assume that K satisfies (5.4.1) , (5.4.2) , and (5.4.3) , and let W be a tempered distribution of the form (5.3.7) that coincides with K on R n ∖{0}. Then we have

(5.4.4)

and consequently

(5.4.5)

Thus the operator given by convolution with W maps L ² ( R n ) to itself with norm at most 15(A ₁ + A ₂ + A ₃ ). Consequently, it also maps L ¹ ( R n ) to L 1,∞ ( R n ) with bound at most a dimensional constant multiple of A ₁ + A ₂ + A ₃ and L p ( R n ) to itself with bound at most $$C_{n}\max (p,(p - 1)^{-1})(A_{1} + A_{2} + A_{3})$$ , when 1 < p < ∞, where C n is a dimensional constant.

Proof.

Let us set $$K^{(\varepsilon,N)}(x) = K(x)\chi _{\varepsilon <\vert x\vert <N}$$. Estimate (5.4.4) implies that for all f in $$\mathcal{S}(\mathbf{R}^{n})$$ we have

uniformly in j. Using this, (5.3.9), and Fatou’s lemma, we obtain that

for all $$f \in \mathcal{S}(\mathbf{R}^{n})$$, and this is equivalent to (5.4.5).

Case 1: Suppose that $$\varepsilon < \vert \xi \vert ^{-1} < N$$. Then we write

where

We now have

(5.4.6)

It follows that

uniformly in $$\varepsilon$$. Let us now examine I ₂(ξ). Let $$z = \frac{\xi } {2\vert \xi \vert ^{2}}$$ so that $$e^{2\pi iz\cdot \xi } = -1$$ and 2 | z |  =  | ξ | ⁻¹. By changing variables x = x′ − z, rewrite I ₂ as

hence averaging gives

Now use that

(5.4.7)

to write $$I_{2}(\xi ) = J_{1}(\xi ) + J_{2}(\xi ) + J_{3}(\xi ) + J_{4}(\xi ) + J_{5}(\xi )$$, where

(5.4.8)

(5.4.9)

(5.4.10)

(5.4.11)

(5.4.12)

Since 2 | z |  =  | ξ | ⁻¹, J ₁(ξ) is bounded in absolute value by $$\frac{1} {2}A_{2}$$, in view of (5.4.2).

Next observe that $$\vert \xi \vert ^{-1} \leq \vert x\vert \leq \frac{3} {2}\vert \xi \vert ^{-1}$$ in (5.4.9), while $$\frac{1} {2}\vert \xi \vert ^{-1} \leq \vert x\vert \leq \vert \xi \vert ^{-1}$$ in (5.4.11); hence both J ₂ and J ₄ are bounded by $$\frac{1} {2}A_{1}$$. Finally, we have $$\frac{1} {2}N < \vert x\vert < N$$ in (5.4.10) (since $$\vert x\vert > N -\frac{1} {2}\vert \xi \vert ^{-1}$$), and similarly we have $$N \leq \vert x\vert < \frac{3} {2}N$$ in (5.4.12). Thus both J ₃ and J ₅ are bounded above by $$\frac{1} {2}A_{1}$$.

Case 2: If $$\varepsilon < N \leq \vert \xi \vert ^{-1}$$, then we write

which is bounded in absolute value by

Notice that if ξ = 0, only the first term appears which is bounded by A ₃.

Case 2: If $$\vert \xi \vert ^{-1} \leq \varepsilon < N$$, we write

and both of the terms on the right are similar to I ₂(ξ) which was shown to be bounded by $$2A_{1} + \frac{1} {2}A_{2}$$.

In all cases (5.4.4) holds. □ 

5.4.2 An Example

We now give an example of a distribution that satisfies conditions (5.4.1), (5.4.2), and (5.4.3).

Example 5.4.2.

Let τ be a nonzero real number and let $$K(x) = \frac{1} {\vert x\vert ^{n+i\tau }}$$ defined for x ∈ R n ∖{0}. For a sequence δ k ↓ 0 and $$\varphi$$ a Schwartz function on R n , define

(5.4.13)

whenever the limit exists. We claim that for some choices of sequences δ k , W is a well defined tempered distribution on R n . Take, for example, $$\delta _{k} = e^{-2\pi k/\tau }$$. For this sequence δ k , observe that

and thus

(5.4.14)

which implies that $$W \in \mathcal{S}'(\mathbf{R}^{n})$$, since

If $$\varphi$$ is supported in R n ∖{0}, then

Therefore W coincides with the function K away from the origin. Moreover, (5.4.1) and (5.4.2) are clearly satisfied for K, while (5.4.3) is also satisfied, since

Remark 5.4.3.

It is important to emphasize that the limit in (5.4.13) may not exist for all sequences δ k  → 0. For example, the limit in (5.4.13) does not exist if $$\delta _{k} = e^{-\pi k/\tau }$$. Moreover, for a different choice of a sequence δ k for which the limit in (5.4.13) exists (for example, $$\delta _{k} = e^{-\pi (2k+1)/\tau }$$), we obtain a different distribution W ₁ that coincides with the function $$K(x) = \vert x\vert ^{-n-i\tau }$$ on R n ∖{0}.

We discuss a point of caution. We can directly check that the distributions W defined by (5.4.13) are not homogeneous distributions of degree − n − i τ. In fact, the only homogeneous distribution of degree − n − i τ that coincides with the function | x | −n−i τ away from zero is a multiple of the distribution u −n−i τ , where u z is defined in (2.​4.​7). Let us investigate the relationship between u −n−i τ and W defined in (5.4.14). Recall that (2.​4.​8) gives

Using (5.4.14), we conclude that $$u_{-n-i\tau } - c_{1}W = c_{2}\delta _{0}$$ for suitable nonzero constants c ₁ and c ₂. Since the Dirac mass at the origin is not a homogeneous distribution of degree − n − i τ, it follows that neither is W.

Since

the identity $$u_{-n-i\tau } - c_{1}W = c_{2}\delta _{0}$$ can be used to obtain a formula for the Fourier transform of W and thus produce a different proof that convolution with W is a bounded operator on L ²(R n ).

5.4.3 Necessity of the Cancellation Condition

Although conditions (5.4.1), (5.4.2), and (5.4.3) are sufficient for L ² boundedness, they might not necessary. However, we show that (5.4.3) is a necessary condition, given (5.4.1).

Proposition 5.4.4.

Suppose that K is a function on R n ∖{0} that satisfies (5.4.1) . Let W be a tempered distribution on R n related to K as in (5.3.7) . If the operator T given by convolution with W maps L ² ( R n ) to itself (equivalently if $$\widehat{W}$$ is an L ∞ function), then the function K must satisfy (5.4.3) .

Proof.

Pick a radial $$\mathcal{C}^{\infty }$$ function $$\varphi$$ supported in the ball | x | ≤ 2 with $$0 \leq \varphi \leq 1$$, and $$\varphi (x) = 1$$ when | x | ≤ 1. For R > 0 let $$\varphi ^{R}(x) =\varphi (x/R)$$. Fourier inversion for distributions gives the second equality,

and the preceding identity implies that

uniformly in R > 0. Fix 0 < R ₁ < R ₂ < ∞. If R ₂ ≤ 2R ₁, we have

which implies the required conclusion. We may therefore assume that 2R ₁ < R ₂. Since the part of the integral in (5.4.3) over the set R ₁ <  | x |  < 2R ₁ is controlled by A ₁, it suffices to control the integral of K(x) over the set 2R ₁ <  | x |  < R ₂. Since the function $$\varphi ^{R_{2}} -\varphi ^{R_{1}}$$ is supported away from the origin, the action of the distribution W on it can be written as integration against the function K. We have

The sum of the last two integrals is bounded by 3A ₁ (since $$0 \leq \varphi \leq 1$$), while the first integral is equal to

and is therefore bounded by $$2\|T\|_{L^{2}\rightarrow L^{2}}\|\widehat{\varphi }\,\|_{L^{1}}$$. We conclude that the function K must satisfy (5.4.3) with constant

This establishes the assertion. □ 

5.4.4 Sufficient Conditions for L p Boundedness of Maximal Singular Integrals

We now discuss the analogous result to Theorem 5.4.1 for the maximal singular integral operator T (∗∗).

Theorem 5.4.5.

Suppose that K satisfies (5.4.1) , (5.4.2) , and (5.4.3) and let T (∗∗) be as in (5.3.18) . Then T (∗∗) is bounded on L p ( R n ), 1 < p < ∞, with norm

where C n is a dimensional constant.

Proof.

We first define an operator T associated with K that satisfies (5.4.1), (5.4.2), and (5.4.3). Because of condition (5.4.3), there exists a sequence $$\delta _{j} \downarrow 0$$ such that

exists. Therefore, for $$\varphi \in \mathcal{S}(\mathbf{R}^{n})$$ we can define a tempered distribution

and an operator T given by T(f) = f ∗ W for $$f \in \mathcal{S}(\mathbf{R}^{n})$$. In view of Theorems 5.4.1 and 5.3.3, T admits an L p bounded extension (1 < p < ∞) with

(5.4.15)

and is weak type (1, 1). This extension is still denoted by T.

Fix 1 < p < ∞ and $$f \in L^{p}(\mathbf{R}^{n}) \cap L^{\infty }(\mathbf{R}^{n})$$ with compact support. We have

where z ₁ and z ₂ are arbitrary points in R n that satisfy $$\vert z_{1} - x\vert \leq \frac{\varepsilon } {2}$$ and $$\vert z_{2} - x\vert \leq \frac{N} {2}$$. We used that f has compact support in order to be able to write $$T^{(\varepsilon )}(f)(x)$$ and T (N)(f)(x) as convergent integrals for almost every x.

At this point we take absolute values, average over $$\vert z_{1} - x\vert \leq \frac{\varepsilon } {2}$$ and $$\vert z_{2} - x\vert \leq \frac{N} {2}$$, and we apply Hölder’s inequality in two terms. We obtain the estimate

where v n is the volume of the unit ball in R n . Applying condition (5.4.2) and estimate (5.4.15), we obtain for f in $$L^{p}(\mathbf{R}^{n}) \cap L^{\infty }(\mathbf{R}^{n})$$ with compact support that

We now use density to remove the compact support condition on f and obtain the last displayed estimate for all functions f in $$L^{p}(\mathbf{R}^{n}) \cap L^{\infty }(\mathbf{R}^{n})$$. Taking the supremum over all $$0 <\varepsilon < N$$ and over all N > 0, we deduce that for all f in $$L^{p}(\mathbf{R}^{n}) \cap L^{\infty }(\mathbf{R}^{n})$$ we have the estimate

(5.4.16)

where S p is the sublinear operator defined by

and M is the Hardy–Littlewood maximal operator.

Recalling that M maps L ¹ to L 1, ∞ with bound at most 3 n and also L p to L p, ∞ with bound at most 2 ⋅ 3 n∕p for 1 < p < ∞ (Exercise 2.1.4), we conclude that S p maps $$L^{p}(\mathbf{R}^{n})$$ to $$L^{p,\infty }(\mathbf{R}^{n})$$ with norm at most

(5.4.17)

where $$\widetilde{c}_{n}$$ is another dimensional constant.

Now write f = f α + f α , where

The function f α is in L ∞ ∩ L p and f α is in L ¹ ∩ L p . Moreover, we see that

(5.4.18)

Apply the Calderón–Zygmund decomposition (Theorem 5.3.1) to the function f α at height α γ to write $$f^{\alpha } = g^{\alpha } + b^{\alpha }$$, where g α is the good function and b α is the bad function of this decomposition. Using (5.3.1), we obtain

(5.4.19)

We now use (5.4.16) to get

(5.4.20)

where

Observe that $$2A_{2}\|f_{\alpha }\|_{L^{\infty }} \leq \alpha /8$$. Selecting $$\gamma = 2^{-n-5}(A_{1} + A_{2})^{-1}$$ and using property (2) in Theorem 5.3.1, we obtain

and therefore

(5.4.21)

Since $$\gamma \leq (2^{n+5}A_{1})^{-1}$$, it follows from (5.3.27) that

and using (5.4.18), we obtain

Using Chebyshev’s inequality in (5.4.21) and (5.4.17), we finally obtain that

Combining the estimates for b ₁, b ₂, and b ₃ and using (5.4.19), we deduce

(5.4.22)

Finally, we need to obtain a similar estimate to (5.4.22), in which the weak L p norm on the left is replaced by the L p norm. This is a consequence of Theorem 1.​3.​2 via interpolation between the estimates $$L^{\frac{p+1} {2} } \rightarrow L^{\frac{p+1} {2},\infty }$$ and $$L^{2p} \rightarrow L^{2p,\infty }$$ for 2 < p < ∞ and between the estimates $$L^{2p} \rightarrow L^{2p,\infty }$$ and L ¹ → L 1, ∞ for 1 < p < 2. The latter estimate follows from Theorem 5.3.5. See also Corollary 5.3.7. □ 

Exercises

5.4.1. Let T be a convolution operator that is L ² bounded. Suppose that a given function $$f_{0} \in L^{1}(\mathbf{R}^{n}) \cap L^{2}(\mathbf{R}^{n})$$ has vanishing integral and that T(f ₀) is integrable. Prove that T(f ₀) also has vanishing integral.

5.4.2. Let K satisfy (5.4.1), (5.4.2), and (5.4.3) and let $$W \in \mathcal{S}'$$ be an extension of K on R n . Let f be a compactly supported $$\mathcal{C}^{1}$$ function on R n with mean value zero. Prove that the function f ∗ W is in L ¹(R n ).

5.4.3. Suppose K is a function on R n ∖{0} that satisfies (5.4.1), (5.4.2), and (5.4.3). Let $$K^{(\varepsilon,N)}(x) = K(x)\chi _{\varepsilon <\vert x\vert <N}$$ for $$0 <\varepsilon < N < \infty $$ and let $$T^{(\varepsilon,N)}$$ be the operator given by convolution with $$K^{(\varepsilon,N)}$$. Let 1 < p < ∞ and $$f \in L^{p}(\mathbf{R}^{n})$$. Prove that for some sequence $$\varepsilon _{j} \downarrow 0$$, $$T^{(\varepsilon _{j},N)}(f)$$ converges almost everywhere as j → ∞ and N → ∞.

[Hint: Use Theorems 5.4.5 and 2.1.14.]

5.4.4. (a) Prove that for all x, y ∈ R n that satisfy 0 ≠ x ≠ y we have

(b) Let Ω be an integrable function with mean value zero on the sphere S n−1. Suppose that Ω satisfies a Lipschitz (Hölder) condition of order 0 < α < 1 on S n−1. This means that

for all θ ₁, θ ₂ ∈ S n−1. Prove that K(x) = Ω(x∕ | x | )∕ | x |  n satisfies Hörmander’s condition (5.4.3) with constant at most a multiple of $$B_{0} +\|\varOmega \| _{L^{\infty }}$$.

5.4.5. Let Ω be an L ¹ function on S n−1 with mean value zero.

(a) Let $$\omega _{\infty }(t) =\sup \{ \vert \varOmega (\theta _{1}) -\varOmega (\theta _{2})\vert:\,\,\theta _{1},\theta _{2} \in \mathbf{S}^{n-1},\,\,\,\vert \theta _{1} -\theta _{2}\vert \leq t\}$$ and suppose that the following Dini condition holds:

Prove that the function $$K(x)\! =\!\varOmega (x/\vert x\vert )\vert x\vert ^{-n}$$ satisfies Hörmander’s condition.

(b) (A. Calderón and A. Zygmund) For A ∈ O(n), let

Suppose that Ω satisfies the more general Dini-type condition

where

Prove the same conclusion as in part (a).

[Hint: Part (b): Use the result in part (a) of Exercise 5.4.4 and switch to polar coordinates.]

5.5 Vector-Valued Inequalities

Certain nonlinear expressions that appear in Fourier analysis, such as maximal functions and square functions, can be viewed as linear quantities taking values in some Banach space. This point of view provides the motivation for a systematic study of Banach-valued operators. Let us illustrate this line of thinking via an example. Let T be a linear operator acting on L p of some measure space (X, μ) and taking values in the set of measurable functions of another measure space (Y, ν). The seemingly nonlinear inequality

(5.5.1)

can be transformed to a linear one with only a slight change of view. Let us denote by L p (X, ℓ ²) the Banach space of all sequences {f j } j of measurable functions on X that satisfy

(5.5.2)

Define a linear operator acting on such sequences by setting

(5.5.3)

Then (5.5.1) is equivalent to the inequality

(5.5.4)

in which $$\mathbf{T}$$ is thought of as a linear operator acting on the L p space of ℓ ²-valued functions on X. This is the basic idea of vector-valued inequalities. A nonlinear inequality such as (5.5.1) can be viewed as a linear norm estimate for an operator acting and taking values in suitable Banach spaces.

5.5.1 ℓ ²-Valued Extensions of Linear Operators

The following result is classical and fundamental in the subject of vector-valued inequalities.

Theorem 5.5.1.

Let 0 < p,q < ∞ and let (X,μ) and (Y,ν) be two σ-finite measure spaces. The following are valid:(a) Suppose that T is a bounded linear operator from L p (X) to L q (Y ) with norm N. Then T has an ℓ ² -valued extension, that is, for all complex-valued functions f j in L p (X) we have

(5.5.5)

for some constant C p,q that satisfies C p,q = 1 if p ≤ q. Moreover, if T maps real-valued L p functions to real-valued L q functions with norm N real , then (5.5.5) holds for real-valued functions f j with N real in place of N.(b) Suppose that T is a bounded linear operator from L p (X) to L q,∞ (Y ) with norm M. Then T has an ℓ ² -valued extension, that is,

(5.5.6)

for some constant D p,q that depends only on p and q. Moreover, if T maps real-valued L p functions to real-valued L q,∞ functions with norm M real , then (5.5.5) holds for real-valued functions f j with M real in place of M.

To prove this theorem, we need the following identities.

Lemma 5.5.2.

For any 0 < r < ∞, define constants

(5.5.7)

Then for any $$\lambda _{1},\lambda _{2},\ldots,\lambda _{n} \in \mathbf{R}$$ we have

(5.5.8)

where dx = dx ₁ ⋯dx n . Also for all $$w_{1},w_{2},\ldots,w_{n} \in \mathbf{C}$$ we have

(5.5.9)

where $$dz = dz_{1}\cdots dz_{n} = dx_{1}dy_{1}\cdots dx_{n}dy_{n}$$ if z j = x j + iy j .

Proof.

Dividing both sides of (5.5.8) by $$(\lambda _{1}^{2} +\ldots +\lambda _{n}^{2})^{\frac{1} {2} }$$, we reduce things to the situation in which $$\lambda _{1}^{2} +\ldots +\lambda _{n}^{2} = 1$$. Let $$e_{1} = (1,0,\ldots,0)^{t}$$ be the standard basis column unit vector on R n and find an orthogonal n × n matrix A ∈ O(n) (orthogonal means a real matrix satisfying A t  = A ⁻¹) such that $$A^{-1}e_{1} = (\lambda _{1},\ldots,\lambda _{n})^{t}$$. Then the first coordinate of A x is

Now change variables y = A x in the integral in (5.5.8) and use the fact that | A x |  =  | x | to obtain

which proves (5.5.8). We used the fact that $$\int _{\mathbf{R}}e^{-\pi \vert x\vert ^{2} }dx = 1$$.

The proof of (5.5.9) is almost identical. We normalize by assuming that

and we let ε ₁ be the column vector of $$\mathbf{C}^{n}$$ having 1 in the first entry and zero elsewhere. We find a unitary n × n matrix $$\mathcal{A}$$ such that $$\mathcal{A}^{-1}\epsilon _{1} = (\overline{w_{1}},\ldots,\overline{w_{n}})^{t}$$. Unitary means $$\mathcal{A}^{-1} = \mathcal{A}^{{\ast}}$$, where $$\mathcal{A}^{{\ast}}$$ is the conjugate transpose matrix of $$\mathcal{A}$$, i.e., the matrix whose entries are the complex conjugates of $$\mathcal{A}^{t}$$ and that satisfies $$u \cdot \overline{\mathcal{A}v} = \mathcal{A}^{{\ast}}u \cdot \overline{v}$$ for all $$u,v \in \mathbf{C}^{n}$$. Then $$(\mathcal{A}z)_{1} = w_{1}z_{1} +\ldots +w_{n}z_{n}$$ and also $$\vert \mathcal{A}z\vert = \vert z\vert $$; therefore, changing variables $$\zeta = \mathcal{A}z$$ in the integral in (5.5.9), we can rewrite that integral as

where the second equality follows by polar coordinates and the third by the change of variables s = t ². □ 

We now continue with the proof of Theorem 5.5.1.

Proof.

The proof is based on conclusion (5.5.9) of Lemma 5.5.2.

Part (a): Assume first that q < p and let B r be as in (5.5.7). We may assume that the sequence {f j } j is indexed by $$\mathbf{Z}^{+}$$. Use successively identity (5.5.9), the boundedness of T, Hölder’s inequality with exponents p∕q and (p∕q)′ with respect to the measure $$e^{-\pi \vert z\vert ^{2} }dz$$, and identity (5.5.9) again to deduce for n ∈ Z ⁺

Now, letting n → ∞ in the previous inequality, we obtain the required conclusion with C p, q  = B p B q ⁻¹ when q < p.

We now turn to the case q ≥ p. Using similar reasoning, we obtain

Note that we made use of Minkowski’s integral inequality (Exercise 1.1.6) in the last inequality. Letting n → ∞ proves the required conclusion with C p, q  = 1 if q ≥ p.

If T happens to map real-valued functions to real-valued functions, then we adapt the preceding argument by taking f j to be real-valued functions, we replace B p by A p , B q by A q , N by N r e a l and we use identity (5.5.8) instead of (5.5.9).

Part (b): Inequality (5.5.6) will be a consequence of (5.5.5) and of the following result of Exercise 1.1.12 which holds when Y is σ-finite:

(5.5.10)

where 0 < r < q and the supremum is taken over all subsets E of Y of finite measure. Using (5.5.10), we obtain

(5.5.11)

where T E is defined by $$T_{E}(f) =\chi _{E}\,T(f)$$. Since for any function f in L p (X) we have

it follows that for any measurable set E of finite measure the estimate

(5.5.12)

is valid. Inserting (5.5.12) in (5.5.11), we obtain (5.5.6) with $$D_{p,q} = C_{p,r}\big( \frac{q} {q-r}\big)^{\!\frac{1} {r} }$$ and 0 < r < q. Recall that C p, r  = 1 if r ≥ p and C p, r  = B p B r ⁻¹ if r < p.

If T happens to map real-valued functions to real-valued functions, then we take f j to be real-valued and we replace M by M r e a l , B r by A r , and B q by A q in the preceding argument. □ 

5.5.2 Applications and ℓ r -Valued Extensions of Linear Operators

Here is an application of Theorem 5.5.1:

Example 5.5.3.

On the real line consider the intervals I j  = [b j , ∞) for j ∈ Z. Let T j be the operator given by multiplication on the Fourier transform by the characteristic function of I j . Then we have the following two inequalities:

(5.5.13)

(5.5.14)

for 1 < p < ∞. To prove these, first observe that the operator $$T = \frac{1} {2}(I + iH)$$ is given on the Fourier transform by multiplication by the characteristic function of the half-axis [0, ∞) [precisely, the Fourier multiplier of T is equal to 1 on the set (0, ∞) and 1∕2 at the origin; this function is almost everywhere equal to the characteristic function of the half-axis [0, ∞)]. Moreover, each T j is given by

and thus with $$g_{j}(x) = e^{-2\pi ib_{j}x}f(x)$$, (5.5.13) and (5.5.14) can be written respectively as

Theorem 5.5.1 gives that both of the previous estimates are valid by in view of the boundedness of $$T = \frac{1} {2}(I + iH)$$ from L p to L p and from $$L^{1} \rightarrow L^{1,\infty }$$. For a slight generalization and an extension to higher dimensions, see Exercise 5.6.1.

We have now seen that bounded operators from L p to L q (or to L q, ∞ ) always admit ℓ ²-valued extensions. It is natural to ask whether they also admit ℓ r -valued extensions for some r ≠ 2. For some values of r we may answer this question. Here is a straightforward corollary of Theorem 5.5.1.

Corollary 5.5.4.

Let (X,μ) and (Y,ν) be σ-finite measure spaces. Suppose that T is a linear bounded operator from L p (X) to L p (Y ) with norm A for some 1 < p < ∞. Let r be a number between p and 2. Then we have

(5.5.15)

Proof.

Using Exercise 5.5.2 we interpolate between the trivial bound $$L^{p}(X,\ell^{p}) \rightarrow L^{p}(Y,\ell^{p})$$ and the bound $$L^{p}(X,\ell^{2}) \rightarrow L^{p}(Y,\ell^{2})$$, which follows from Theorem 5.5.1. We obtain the bound $$L^{p}(X,\ell^{r}) \rightarrow L^{p}(Y,\ell^{r})$$ since r lies between p and 2. □ 

Example 5.5.5.

The result of Corollary 5.5.4 may fail if r does not lie in the interval with endpoints p and 2. Let us take, for example, 1 < p < 2 and consider an r < p. Take X = Y = R and define a linear operator T by setting

Then T is L p bounded, since $$\|T(f)\|_{L^{p}} \leq \| T(f)\|_{L^{p'}} \leq \| f\|_{L^{p}}$$. Now take f j  = χ [j−1, j] for $$j = 1,\ldots,N$$. A simple calculation gives

while

It follows that $$N^{1/r} \leq CN^{1/p}$$ for all N > 1, and hence (5.5.15) cannot hold if p > r.

We have now seen that ℓ r -valued extensions for r ≠ 2 may fail in general. But do they fail for some specific operators of interest in Fourier analysis? For instance, is the inequality

(5.5.16)

true for the Hilbert transform H whenever 1 < p, r < ∞? The answer to this question is affirmative. Inequality (5.5.16) is indeed valid and was first proved using complex function theory. In the next section we plan to study inequalities such as (5.5.16) for general singular integrals using the Calderón–Zygmund theory of the previous section applied to the context of Banach-valued functions.

5.5.3 General Banach-Valued Extensions

Let $$\mathcal{B}$$ be a Banach space over the field of complex numbers with norm $$\|\,\,\,\,\|_{\mathcal{B}}$$, and let $$\mathcal{B}^{{\ast}}$$ be its dual (with norm $$\|\,\,\,\,\|_{\mathcal{B}^{{\ast}}}$$). A function F defined on a measure space (X, μ) and taking values in $$\mathcal{B}$$ is called $$\mathcal{B}$$-measurable if there exists a measurable subset X ₀ of X such that μ(X ∖ X ₀) = 0, F[X ₀] is contained in some separable subspace $$\mathcal{B}_{0}$$ of $$\mathcal{B}$$, and for every $$u^{{\ast}}\in \mathcal{B}^{{\ast}}$$ the complex-valued map

is measurable. A consequence of this definition is that the positive function $$x\mapsto \|F(x)\|_{\mathcal{B}}$$ on X is measurable; to see this, use the relevant result in [382, p. 131].

For 0 < p ≤ ∞ we denote by L p (X) the space L p (X, C). Let $$L^{p}(X) \otimes \mathcal{B}$$ be the set of all finite linear combinations of elements of $$\mathcal{B}$$ with coefficients in L p (X), that is, elements of the form

(5.5.17)

where f j  ∈ L p (X), $$u_{j} \in \mathcal{B}$$, and m ∈ Z ⁺. We define $$L^{p}(X,\mathcal{B})$$ to be the space of all $$\mathcal{B}$$-measurable functions F on X satisfying

(5.5.18)

with the obvious modification when p = ∞. Similarly define $$L^{p,\infty }(X,\mathcal{B})$$ as the space of all $$\mathcal{B}$$-measurable functions F on X satisfying

(5.5.19)

Then $$L^{p}(X,\mathcal{B})$$ (respectively, $$L^{p,\infty }(X,\mathcal{B})$$) is called the L p (respectively, L p, ∞ ) space of functions on X with values in $$\mathcal{B}$$. The quantity in (5.5.18) (respectively, in (5.5.19)) is the norm of F in $$L^{p}(X,\mathcal{B})$$ (respectively, in $$L^{p,\infty }(X,\mathcal{B})$$).

Proposition 5.5.6.

Let $$\mathcal{B}$$ be a Banach space and (X,μ) a σ-finite measure space.

(a) The set $$\{\sum _{j=1}^{m}\chi _{E_{j}}\,u_{j}:\, \mbox{ $u_{j} \in \mathcal{B}$, $E_{j} \subseteq X$ are pairwise disjoint and $\mu (E_{j}) < \infty $}\}$$ is dense in $$L^{p}(X,\mathcal{B})$$ whenever 0 < p < ∞.

(b) The set $$\{\sum _{j=0}^{\infty }\chi _{E_{j}}\,u_{j}:\, \mbox{ $u_{j} \in \mathcal{B}$, $E_{j} \subseteq X$ are pairwise disjoint and $X = \cup _{j=0}^{\infty }E_{j}$}\}$$ is dense in $$L^{\infty }(X,\mathcal{B})$$ .

(c) The space $$\mathcal{C}_{0}^{\infty }\otimes \mathcal{B}$$ of functions of the form $$\sum _{j=1}^{m}\varphi _{j}\,u_{j}$$ , where $$u_{j} \in \mathcal{B}$$,$$\varphi _{j}$$ are in $$\mathcal{C}_{0}^{\infty }(\mathbf{R}^{n})$$ , is dense in $$L^{p}(\mathbf{R}^{n},\mathcal{B})$$ for $$1 \leq p < \infty $$ .

Proof.

If $$F \in L^{p}(X,\mathcal{B})$$ for 0 < p ≤ ∞, then F is $$\mathcal{B}$$-measurable; thus there exists $$X_{0} \subseteq X$$ satisfying $$\mu (X\setminus X_{0}) = 0$$ and $$F[X_{0}] \subseteq \mathcal{B}_{0}$$, where $$\mathcal{B}_{0}$$ is some separable subspace of $$\mathcal{B}$$. Choose a countable dense sequence $$\{u_{j}\}_{j=1}^{\infty }$$ of $$\mathcal{B}_{0}$$.

(a) First assume that p < ∞. Since X is σ-finite, for any $$\varepsilon > 0$$, there exists a measurable subset X ₁ of X ₀ with μ(X ₁) < ∞ such that

Setting

we have $$\mathcal{B}_{0} \subseteq \bigcup _{j=1}^{\infty }\widetilde{B}(u_{j},\varepsilon )$$. Let $$A_{1} =\widetilde{ B}(u_{1},\varepsilon )$$ and $$A_{j} =\widetilde{ B}(u_{j},\varepsilon )\setminus (\bigcup _{i=1}^{j-1}\widetilde{B}(u_{i},\varepsilon ))$$ for j ≥ 2. It is easily seen that $$\{A_{j}\}_{j=1}^{\infty }$$ are pairwise disjoint and $$\bigcup _{j=1}^{\infty }A_{j} =\bigcup _{ j=1}^{\infty }\widetilde{B}(u_{j},\varepsilon )$$. Set $$E_{j} = F^{-1}[A_{j}] \cap X_{1}$$. Then $$X_{1} =\bigcup _{ j=1}^{\infty }E_{j}$$ and $$\{E_{j}\}_{j=1}^{\infty }$$ are pairwise disjoint. Since $$\mu (X_{1}) =\sum _{ j=1}^{\infty }\mu (E_{j}) < \infty $$, it follows that μ(E j ) < ∞ and also that for some m ∈ Z ⁺,

(5.5.20)

Moreover, one can easily verify that $$\sum _{j=1}^{m}\chi _{E_{j}}u_{j}$$ is $$\mathcal{B}$$-measurable. Notice that $$\|F(x) - u_{j}\|_{\mathcal{B}} <\varepsilon (3\mu (X_{1}))^{-1/p}$$ for any x ∈ E j and $$j \in \{ 1,\ldots,m\}$$. This fact combined with (5.5.20) and the mutual disjointness of {E j } j = 1 m yields that

(b) Now consider the case p = ∞. Obviously we have $$\mathcal{B}_{0} \subseteq \bigcup _{j=1}^{\infty }B(u_{j},\varepsilon )$$, where $$B(u_{j},\varepsilon ) =\{ u \in \mathcal{B}_{0}:\| u - u_{j}\|_{\mathcal{B}} <\varepsilon \}$$. Let $$A_{1} = B(u_{1},\varepsilon )$$ and for j ≥ 2 define sets $$A_{j} = B(u_{j},\varepsilon )\setminus (\bigcup _{i=1}^{j-1}B(u_{i},\varepsilon ))$$. Let E j  = F ⁻¹[A j ] for j ≥ 1 and $$E_{0} = X\setminus (\bigcup _{j=1}^{\infty }E_{j})$$. Then μ(E ₀) = 0. As in the proof of the case p < ∞, we have that $$\{E_{j}\}_{j=0}^{\infty }$$ are pairwise disjoint and $$X_{0} \subseteq \bigcup _{j=0}^{\infty }E_{j}$$. Pick u ₀ = 0. Notice that $$\sum _{j=0}^{\infty }\chi _{E_{j}}u_{j}$$ is $$\mathcal{B}$$-measurable. Since $$\|F(x) - u_{j}\|_{\mathcal{B}} <\varepsilon$$ for any x ∈ E j and j ≥ 0, we have

which completes the proof in the case p = ∞.

(c) For the last assertion, we fix a smooth function with supported in the unit ball of R n with integral one. Let $$\varphi _{\delta }(x) =\delta ^{-n}\varphi (x/\delta )$$ for x ∈ R n and $$\delta > 0$$. Given a function $$\sum _{j=1}^{m}\chi _{E_{j}}\,u_{j}$$ as in part (a) approximating a given f in $$L^{p} \otimes \mathcal{B}$$, we consider the function $$\sum _{j=1}^{m}(\chi _{E_{j}} {\ast}\varphi _{\delta })\,u_{j}$$ which lies in $$\mathcal{C}_{0}^{\infty }\otimes \mathcal{B}$$. Since $$\|\chi _{E_{j}} {\ast}\varphi _{\delta }-\chi _{E_{j}}\|_{L^{p}} \rightarrow 0$$ as δ → 0 when 1 ≤ p < ∞, $$\sum _{j=1}^{m}(\chi _{E_{j}} {\ast}\varphi _{\delta })\,u_{j}$$ tends to $$\sum _{j=1}^{m}\chi _{E_{j}}\,u_{j}$$ in $$L^{p} \otimes \mathcal{B}$$ as $$\delta \rightarrow 0$$, and the conclusion follows. □ 

Let (X, μ) be a measure space. If F is an element of $$L^{1}(X) \otimes \mathcal{B}$$ given as in (5.5.17), we define its integral (which is an element of $$\mathcal{B}$$) by setting

Observe that for every $$F \in L^{1}(X) \otimes \mathcal{B}$$ we have

Thus the linear operator

is bounded from $$L^{1}(X) \otimes \mathcal{B}$$ into $$\mathcal{B}$$. Since every element of $$L^{1}(X,\mathcal{B})$$ is a (norm) limit (Proposition 5.5.6 (c)) of a sequence of elements in $$L^{1}(X) \otimes \mathcal{B}$$, by continuity, the operator F↦I F has a unique extension on $$L^{1}(X,\mathcal{B})$$ that we call the Bochner integral of F and denote by

$$L^{1}(X,\mathcal{B})$$ is called the space of all Bochner integrable functions from X to $$\mathcal{B}$$. Since the Bochner integral is an extension of I F , for each $$F \in L^{1}(X,\mathcal{B})$$ we have

Consequently, measurable functions F with $$\int _{X}\|F(x)\|_{\mathcal{B}}\,dx < \infty $$ are Bochner integrable over X. It is not difficult to show that the Bochner integral of F is the only element of $$\mathcal{B}$$ that satisfies

(5.5.21)

for all $$u^{{\ast}}\in \mathcal{B}^{{\ast}}$$. The next result concerns duality in this context when X = R n .

Proposition 5.5.7.

Let $$\mathcal{B}$$ be a Banach space and $$1 \leq p \leq \infty $$ .

(a) For any $$F \in L^{p}(\mathbf{R}^{n},\mathcal{B})$$ we have

Consequently,$$L^{p}(\mathbf{R}^{n},\mathcal{B})$$ isometrically embeds in $$(L^{p'}(\mathbf{R}^{n},\mathcal{B}^{{\ast}}))^{{\ast}}$$ .

(b) for any $$G \in L^{p'}(\mathbf{R}^{n},\mathcal{B}^{{\ast}})$$ one has

and thus $$L^{p'}(\mathbf{R}^{n},\mathcal{B}^{{\ast}})$$ isometrically embeds in $$(L^{p}(\mathbf{R}^{n},\mathcal{B}))^{{\ast}}$$ .

Proof.

Hölder’s inequality yields that the right-hand side of (a) is controlled by its left-hand side. It remains to establish the reverse inequality.

For $$F \in L^{p}(\mathbf{R}^{n},\mathcal{B})$$ and $$\varepsilon > 0$$, by Proposition 5.5.6, there is $$F_{\varepsilon }(x)\! =\!\sum _{ j=1}^{m}\chi _{E_{j}}(x)u_{j}$$ with m ∈ Z ⁺ or m = ∞ (when p = ∞) such that $$\|F_{\varepsilon } - F\|_{L^{p}(\mathbf{R}^{n},\,\mathcal{B})} <\varepsilon /2$$, where {E j } j = 1 m are pairwise disjoint subsets of R n and $$u_{j} \in \mathcal{B}$$. Since $$F_{\varepsilon } \in L^{p}(\mathbf{R}^{n},\mathcal{B})$$, we choose a nonnegative function h satisfying $$\|h\|_{L^{p'}(\mathbf{R}^{n})} \leq 1$$ such that

(5.5.22)

When 1 ≤ p < ∞, we can further choose $$h \in L^{p'}(\mathbf{R}^{n})$$ to be a function with bounded support, which ensures that it is integrable. For given $$u_{j} \in \mathcal{B}$$, there exists $$u_{j}^{{\ast}}\in \mathcal{B}^{{\ast}}$$ satisfying $$\|u_{j}^{{\ast}}\|_{\mathcal{B}^{{\ast}}} = 1$$ and

(5.5.23)

Set $$G(x) =\sum _{ j=1}^{m}h(x)\chi _{E_{j}}(x)u_{j}^{{\ast}}$$. Clearly G is $$\mathcal{B}^{{\ast}}$$-measurable and $$\|G\|_{L^{p'}(\mathbf{R}^{n},\,\mathcal{B}^{{\ast}})} \leq 1$$. It follows from (5.5.22) and (5.5.23) that

Using Hölder’s inequality we have

hence we obtain

Letting $$\varepsilon \rightarrow 0$$ yields the desired inequality in part (a).

(b) The duality statement $$\vert \langle G(x),F(x)\rangle \vert \leq \| G(x)\|_{\mathcal{B}^{{\ast}}}\|F(x)\|_{\mathcal{B}}$$ together with Hölder’s inequality imply the ≥ inequality in part (b). We prove the ≤ inequality via an argument symmetric with that in case (a). For completeness we include the details. For a given G in $$L^{p'}(\mathbf{R}^{n},\mathcal{B})$$ and $$\varepsilon > 0$$, by Proposition 5.5.6, there is $$G_{\varepsilon }(x)\! =\!\sum _{ j=1}^{m}\chi _{E_{j}}(x)u_{j}^{{\ast}}$$ with m ∈ Z ⁺ or m = ∞ (when p = 1) such that $$\|G_{\varepsilon } - G\|_{L^{p'}(\mathbf{R}^{n},\,\mathcal{B}^{{\ast}})} <\varepsilon /2$$, where $$\{E_{j}\}_{j=1}^{m}$$ are pairwise disjoint subsets of R n and $$u_{j}^{{\ast}}\in \mathcal{B}^{{\ast}}$$. Since $$G_{\varepsilon }$$ lies in $$L^{p'}(\mathbf{R}^{n},\mathcal{B}^{{\ast}})$$, we choose a nonnegative function h satisfying $$\|h\|_{L^{p}(\mathbf{R}^{n})} \leq 1$$ such that

(5.5.24)

When $$1 < p \leq \infty $$, we can further choose $$h \in L^{p}(\mathbf{R}^{n})$$ to be a function with bounded support, which ensures that it is integrable. For each $$u_{j}^{{\ast}}\in \mathcal{B}$$, there exists $$u_{j} \in \mathcal{B}$$ satisfying $$\|u_{j}\|_{\mathcal{B}} = 1$$ and

(5.5.25)

Set $$F(x) =\sum _{ j=1}^{m}h(x)\chi _{E_{j}}(x)u_{j}$$. Clearly F is $$\mathcal{B}$$-measurable and $$\|F\|_{L^{p}(\mathbf{R}^{n},\,\mathcal{B})} \leq 1$$. It follows from (5.5.24) and (5.5.25) that

Hence, for any $$\varepsilon > 0$$, we have

which implies the reverse inequality in part (b) by letting $$\varepsilon \rightarrow 0$$. □ 

Definition 5.5.8.

Let X, Y be measure spaces. Let T be a linear operator that maps L p (X) to L q (Y ) (respectively, L p (X) to L q, ∞ (Y )) for some 0 < p, q ≤ ∞. We define another operator $$\mathbf{T}$$ acting on $$L^{p}(X) \otimes \mathcal{B}$$ by setting

If $$\mathbf{T}$$ happens to have a bounded extension from $$L^{p}(X,\mathcal{B})$$ to $$L^{q}(Y,\mathcal{B})$$ (respectively from $$L^{p}(X,\mathcal{B})$$ to $$L^{q,\infty }(Y,\mathcal{B})$$), then we say that T has a bounded $$\mathcal{B}$$ -valued extension. In this case we also denote by $$\mathbf{T}$$ the $$\mathcal{B}$$-valued extension of T.

Example 5.5.9.

Let $$\mathcal{B} =\ell ^{r}$$ for some $$1 \leq r < \infty $$. Then a measurable function $$F:\, X \rightarrow \mathcal{B}$$ is just a sequence {f j } j of measurable functions $$f_{j}:\, X \rightarrow \mathbf{C}$$. The space $$L^{p}(X,\ell^{r})$$ consists of all measurable complex-valued sequences {f j } j on X that satisfy

The space $$L^{p}(X) \otimes \ell^{r}$$ is the set of all finite sums

where $$g_{j} \in L^{p}(X)$$ and $$(a_{j1},a_{j2},a_{j3},\ldots ) \in \ell^{r}$$, $$j = 1,\ldots,m$$. This is certainly a subspace of L p (X, ℓ r ). Now given $$(f_{1},f_{2},\ldots ) \in L^{p}(X,\ell^{r})$$, let $$F_{m} = e_{1}f_{1} +\ldots +e_{m}f_{m}$$, where e j is the infinite sequence with zeros everywhere except at the jth entry, where it has 1. Then $$F_{m} \in L^{p}(X) \otimes \ell^{r}$$ and approximates f in the norm of L p (X, ℓ r ). This shows the density of L p (X) ⊗ ℓ r in L p (X, ℓ r ).

If T is a linear operator bounded from L p (X) to L q (Y ), then $$\mathbf{T}$$ is defined by

According to Definition 5.5.8, T has a bounded ℓ r -extension if and only if the inequality

is valid.

A linear operator T acting on measurable functions is called positive if it satisfies $$f \geq 0\Rightarrow T(f) \geq 0$$. It is straightforward to verify that positive operators satisfy

(5.5.26)

for all f, g, f j measurable functions. We have the following result regarding vector-valued extensions of positive operators:

Proposition 5.5.10.

Let $$0 < p,q \leq \infty $$ and (X,μ), (Y,ν) be two measure spaces. Let T be a positive linear operator mapping L p (X) to L q (Y ) (respectively, to L q,∞ (Y )) with norm A. Let $$\mathcal{B}$$ be a Banach space. Then T has a $$\mathcal{B}$$ -valued extension $$\mathbf{T}$$ that maps $$L^{p}(X,\mathcal{B})$$ to $$L^{q}(Y,\mathcal{B})$$ (respectively, to $$L^{q,\infty }(Y,\mathcal{B})$$ ) with the same norm.

Proof.

Let us first understand this theorem when $$\mathcal{B} =\ell ^{r}$$ for 1 ≤ r ≤ ∞. The two endpoint cases r = 1 and r = ∞ can be checked easily using the properties in (5.5.26). For instance, for r = 1 we have

while for r = ∞ we have

The required inequality for 1 < r < ∞,

follows from the Riesz–Thorin interpolation theorem (see Exercise 5.5.2).

The result for a general Banach space $$\mathcal{B}$$ can be proved using the following inequality:

(5.5.27)

by simply taking L q norms. To prove (5.5.27), let us take $$F =\sum _{ j=1}^{n}f_{j}u_{j}$$. Then

where the inequality makes use of the fact that T is a positive operator. □ 

Exercises

5.5.1. ([207]) Let (X, μ) and (Y, ν) be σ-finite measure spaces and suppose that 0 < p ₁, p ₁ ≤ ∞, 1 ≤ q ₀, q ₁ ≤ ∞, and that p ₀ > p ₁. For 0 < θ < 1 define p, q by

Let $$\mathcal{B}_{1}$$ and $$\mathcal{B}_{2}$$ be Banach spaces and let $$\mathbf{T}$$ be a linear operator that maps $$L^{p_{0}}(X,\mathcal{B}_{1})$$ to $$L^{q_{0}}(Y,\mathcal{B}_{2})$$ with norm A ₀ and $$L^{p_{1}}(X,\mathcal{B}_{1})$$ to $$L^{q_{1}}(Y,\mathcal{B}_{2})$$ with norm A ₁. Show that $$\mathbf{T}$$ has an extension that maps $$L^{p}(X,\mathcal{B}_{1})$$ to $$L^{q}(Y,\mathcal{B}_{2})$$ with norm at most $$9\,A_{0}^{1-\theta }A_{1}^{\theta }$$, by following the steps below:

(a) Let i ∈ { 0, 1}. If F j , $$j = 1,\ldots,m$$ are in $$L^{p_{i}}(\mathcal{B}_{1})$$ with disjoint supports, using for y ∈ Y show that

(b) Assume that F lies in a dense subspace of $$\mathcal{B}_{1}$$, it satisfies $$\|\|F\|_{\mathcal{B}_{1}}\|_{L^{p}} = 1$$, and it takes only finitely many values. For a λ > 1 pick a large integer N such that $$\lambda ^{-N} <\| F(x)\|_{\mathcal{B}_{1}} \leq \lambda ^{N}$$ for all x ∈ X such that $$\|F(x)\|_{\mathcal{B}_{1}}\neq 0$$ and define $$F_{j} = F\chi _{\varOmega _{j}}$$, where $$\varOmega _{j} =\{ x:\,\, 2^{j} <\| F\|_{\mathcal{B}_{1}} \leq 2^{j+1}\}$$. Let $$a = \frac{p} {p_{1}} - \frac{p} {p_{0}}$$. Prove the inequalities

(c) Define $$g_{0}(y) =\max _{j}\lambda ^{-ja\theta }\|\mathbf{T}(F_{j})(y)\|_{\mathcal{B}_{2}}$$, $$g_{1}(y) =\max _{j}\lambda ^{ja(1-\theta )}\|\mathbf{T}(F_{j})(y)\|_{\mathcal{B}_{2}}$$ for y ∈ Y and show that

(d) Prove that for all y ∈ Y we have

and conclude that $$\|\mathbf{T}(F)\|_{L^{p}(Y,\mathcal{B}_{2})} \leq 9\,A_{0}^{1-\theta }A_{1}^{\theta }$$ by picking $$\lambda = (1 + \sqrt{2})^{a\theta }$$.

[Hint: Part (d): Split the sum according to whether $$\lambda ^{ja} > \frac{g_{1}(y)} {g_{0}(y)}$$ and $$\lambda ^{ja} \leq \frac{g_{1}(y)} {g_{0}(y)}$$.]

5.5.2. Prove the following version of the Riesz–Thorin interpolation theorem. Let (X, μ) and (Y, ν) be σ-finite measure spaces. Let $$1 < p_{0},q_{0},,p_{1},q_{1},r_{0},s_{0},r_{1},s_{1} < \infty $$ and 0 < θ < 1 satisfy

Suppose that T is a linear operator that maps $$L^{p_{0}}(X)$$ to $$L^{q_{0}}(Y )$$ and $$L^{p_{1}}(X)$$ to $$L^{q_{1}}(Y )$$. Define a vector-valued operator $$\mathbf{T}$$ by setting $$\mathbf{T}(\{f_{j}\}_{j}) =\{ T(f_{j})\}_{j}$$ acting on sequences of complex-valued functions defined on X. Suppose that $$\mathbf{T}$$ maps $$L^{p_{0}}(X,\ell^{r_{0}}(\mathbf{C}))$$ to $$L^{q_{0}}(Y,\ell^{s_{0}}(\mathbf{C}))$$ with norm M ₀ and $$L^{p_{1}}(X,\ell^{r_{1}}(\mathbf{C}))$$ to $$L^{q_{1}}(Y,\ell^{s_{1}}(\mathbf{C}))$$ with norm M ₁. Prove that $$\mathbf{T}$$ maps $$L^{p}(X,\ell^{r}(\mathbf{C}))$$ to $$L^{q}(Y,\ell^{s}(\mathbf{C}))$$ with norm at most $$M_{0}^{1-\theta }M_{1}^{\theta }$$.

[Hint: Use the idea of the proof of Theorem 1.​3.​4.​ Apply Lemma 1.​3.​5 to the function

where $$P(z) = \frac{p} {p_{0}} (1 - z) + \frac{p} {p_{1}} z$$, $$Q(z) = \frac{q'} {q_{0}'} (1 - z) + \frac{q'} {q_{1}'} z\,$$, $$R(z) = \frac{r} {r_{0}} (1 - z) + \frac{r} {r_{1}} z$$, and $$S(z) = \frac{s'} {s_{0}'} (1 - z) + \frac{s'} {s_{1}'} z$$, $$\{a_{k,l}\}_{l\in \mathbf{Z}},\{b_{j,l}\}_{l\in \mathbf{Z}}$$ are finitely supported sequences of positive reals, $$\alpha _{k,l},\beta _{k,l} \in \mathbf{R}$$, A k are pairwise disjoint subsets of X with finite measure, and B j are pairwise disjoint subsets of Y with finite measure.]

5.5.3. Prove the following version of the Marcinkiewicz interpolation theorem. Let (X, μ) and (Y, ν) be σ-finite measure spaces and let 0 < p ₀ < p < p ₁ ≤ ∞ and 0 < θ < 1 satisfy

Let $$\mathcal{B}_{1},\mathcal{B}_{2}$$ be Banach spaces and $$\mathbf{T}$$ be defined on $$L^{p_{0}}(X,\mathcal{B}_{1}) + L^{p_{1}}(X,\mathcal{B}_{1})$$ such that for every y ∈ Y and for all $$F,G \in L^{p_{0}}(X,\mathcal{B}_{1}) + L^{p_{1}}(X,\mathcal{B}_{1})$$ we have

(a) Suppose that $$\mathbf{T}$$ maps $$L^{p_{0}}(X,\mathcal{B}_{1})$$ to $$L^{p_{0},\infty }(Y,\mathcal{B}_{2})$$ with norm A ₀ and $$L^{p_{1}}(X,\mathcal{B}_{1})$$ to $$L^{p_{1},\infty }(Y,\mathcal{B}_{2})$$ with norm A ₁. Show that $$\mathbf{T}$$ maps $$L^{p}(X,\mathcal{B}_{1})$$ to $$L^{p}(Y,\mathcal{B}_{2})$$ with norm at most $$2\big( \frac{p} {p-p_{0}} + \frac{p} {p_{1}-p}\big)^{\frac{1} {p} }A_{0}^{1-\theta }A_{1}^{\theta }$$.

(b) Let p ₀ = 1. If $$\mathbf{T}$$ is linear and maps $$L^{1}(X,\mathcal{B}_{1})$$ to $$L^{1,\infty }(Y,\mathcal{B}_{2})$$ with norm A ₀ and $$L^{p_{1}}(X,\mathcal{B}_{1})$$ to $$L^{p_{1}}(Y,\mathcal{B}_{2})$$ with norm A ₁, show that $$\mathbf{T}$$ maps $$L^{p}(X,\mathcal{B}_{1})$$ to $$L^{p}(Y,\mathcal{B}_{2})$$ with norm at most $$72\big(p - 1\big)^{-1/p}A_{0}^{1-\theta }A_{1}^{\theta }$$.

[Hint: Part (a): Copy the proof of Theorem 1.​3.​2.​ Part (b): Use Exercise 5.5.1.]

5.5.4. For all x ∈ R n let $$\mathbf{K}(x)$$ be a bounded linear operator from $$\mathcal{B}_{1}$$ to $$\mathcal{B}_{2}$$ and let $$\mathcal{Q}\otimes \mathcal{B}_{1}$$ the space of all finite linear combinations of elements of the form $$F =\sum _{ i=1}^{m}\chi _{E_{i}}u_{i}\,,$$ where E i are disjoint measurable subsets of R n of finite measure, u i in $$\mathcal{B}_{1}$$, and $$m \in \mathbf{Z}^{+}$$.

(a) Suppose that $$\mathbf{K}$$ satisfies

Prove that the operator

initially defined on $$\mathcal{Q}\otimes \mathcal{B}_{1}$$, has an extension that maps $$L^{p}(\mathbf{R}^{n},\mathcal{B}_{1})$$ to $$L^{p}(\mathbf{R}^{n},\mathcal{B}_{2})$$ with norm at most C ₁ for 1 ≤ p < ∞.

(b) (Young’s inequality) Let $$1 \leq p,q,s \leq \infty $$ be such that p < ∞, s > 1, 1∕q + 1 = 1∕s + 1∕p. Suppose that $$\mathbf{K}$$ satisfies

Prove that the $$\mathbf{T}$$ defined in part (a) has an extension that maps $$L^{p}(\mathbf{R}^{n},\mathcal{B}_{1})$$ to $$L^{q}(\mathbf{R}^{n},\mathcal{B}_{2})$$ with norm at most C ₂.

(c) (Young’s inequality for weak type spaces) Suppose that 1 < p, s < ∞, 1∕q + 1 = 1∕s + 1∕p, and that $$\mathbf{K}$$ satisfies

Prove that the $$\mathbf{T}$$ defined in part (a) has an extension that maps $$L^{p}(\mathbf{R}^{n},\mathcal{B}_{1})$$ to $$L^{q}(\mathbf{R}^{n},\mathcal{B}_{2})$$.

(d) Prove the following (slight) generalization of the assertion in part (a). Suppose that $$\mathbf{K}$$ satisfies

for all $$u \in \mathcal{B}_{1}$$. Then $$\mathbf{T}$$ has an extension that maps $$L^{1}(\mathbf{R}^{n},\mathcal{B}_{1})$$ to $$L^{1}(\mathbf{R}^{n},\mathcal{B}_{2})$$ with norm at most C ₁.

5.5.5. Use the inequality for the Rademacher functions in Appendix C.2 instead of Lemma 5.5.2 to prove part (a) of Theorem 5.5.1 in the special case p = q. Notice that this approach does not yield a sharp constant.

5.5.6. Let $$0 < p\neq 2 \leq \infty $$ and suppose that T j are uniformly bounded linear operators from L p (R) to L p (R). Show that the inequality

may fail.

[Hint: Let $$T_{j}(g)(x) = g(x - j)$$. When p > 2 take f j (x) = χ [−j, 1−j] for $$j = 1,2,\ldots,N$$. When p < 2 take f j  = χ [0, 1] for $$j = 1,2,\ldots,N$$.]

5.5.7. Suppose that T is a linear operator that takes real-valued functions to real-valued functions. Use Theorem 5.5.1(a) with p = q to prove that

5.6 Vector-Valued Singular Integrals

We now discuss some results about vector-valued singular integrals. By this we mean singular integral operators acting on functions defined on R n and taking values in Banach spaces.

5.6.1 Banach-Valued Singular Integral Operators

Suppose that $$\mathcal{B}_{1}$$ and $$\mathcal{B}_{2}$$ are Banach spaces. We denote by $$L(\mathcal{B}_{1},\mathcal{B}_{2})$$ the space of all bounded linear operators from $$\mathcal{B}_{1}$$ to $$\mathcal{B}_{2}$$. We consider a kernel $$\mathbf{K}$$ defined on R n ∖{0} that takes values in $$L(\mathcal{B}_{1},\mathcal{B}_{2})$$. In other words, for all x ∈ R n ∖{0}, $$\mathbf{K}(x)$$ is a bounded linear operator from $$\mathcal{B}_{1}$$ to $$\mathcal{B}_{2}$$ with norm $$\|\mathbf{K}(x)\|_{\mathcal{B}_{1}\rightarrow \mathcal{B}_{2}}$$. Thus for any $$v \in \mathcal{B}_{1}$$ and any $$x \in \mathbf{R}^{n}\setminus \{0\}$$ we have

We assume that there is a constant A < ∞ such that the size condition holds

(5.6.1)

and also the regularity condition

(5.6.2)

Moreover, we assume that there is a sequence $$\varepsilon _{k} \downarrow 0$$ as k → ∞ and an element $${\boldsymbol K_{0}}$$ of $$L(\mathcal{B}_{1},\mathcal{B}_{2})$$ such that

(5.6.3)

Given these assumptions, we define an operator $$\mathbf{T}$$ on $$\mathcal{C}_{0}^{\infty }\otimes \mathcal{B}_{1}$$ as follows:

For functions $$f_{i} \in \mathcal{C}_{0}^{\infty }(\mathbf{R}^{n})$$ and $$u_{i} \in \mathcal{B}_{1}$$ we define

(5.6.4)

(5.6.5)

Notice that for each $$i \in \{ 1,\ldots,m\}$$ we have

and this is a finite integral in view of (5.6.1). Thus the function

is $$\mathcal{B}_{2}$$-integrable and the expression

is a well-defined element of $$\mathcal{B}_{2}$$. Also the integral in (5.6.5) is over the compact set 1 ≤ | y | ≤ | x | + M, where the ball B(0, M) contains the supports of all f i , and thus it also converges in $$\mathcal{B}_{2}$$, using (5.6.1).

The following vector-valued extension of Theorem 5.3.3 is the main result of this section.

Theorem 5.6.1.

Let $$\mathcal{B}_{1}$$ and $$\mathcal{B}_{2}$$ be Banach spaces. Suppose that $$\mathbf{K}(x)$$ satisfies (5.6.1), (5.6.2) , and (5.6.3) for some A > 0 and $${\boldsymbol K_{0}} \in L(\mathcal{B}_{1},\mathcal{B}_{2})$$ . Let $$\mathbf{T}$$ be the operator associated with $$\mathbf{K}$$ as in (5.6.4) . Assume that $$\mathbf{T}$$ is a bounded linear operator from $$L^{r}(\mathbf{R}^{n},\mathcal{B}_{1})$$ to $$L^{r}(\mathbf{R}^{n},\mathcal{B}_{2})$$ with norm $$B_{\star }$$ for some $$1 < r \leq \infty $$ . Then $$\mathbf{T}$$ has well-defined extensions on $$L^{p}(\mathbf{R}^{n},\mathcal{B}_{1})$$ for all 1 ≤ p < ∞. Moreover, there exist dimensional constants C n and C n ′ such that

(5.6.6)

for all F in $$L^{1}(\mathbf{R}^{n},\mathcal{B}_{1})$$ and

(5.6.7)

whenever 1 < p < ∞ and F is in $$L^{p}(\mathbf{R}^{n},\mathcal{B}_{1})$$ .

Proof.

Although $$\mathbf{T}$$ is defined on the entire $$L^{1}(\mathbf{R}^{n},\mathcal{B}_{1}) \cap L^{r}(\mathbf{R}^{n},\mathcal{B}_{1})$$, it will be convenient to work with its restriction to a smaller dense subspace of $$L^{1}(\mathbf{R}^{n},\mathcal{B}_{1})$$. We make the observation that the space $$\mathcal{Q}\otimes \mathcal{B}_{1}$$ of all functions of the form $$\sum _{i=1}^{m}\chi _{R_{i}}u_{i}$$, where R i are disjoint dyadic cubes and $$u_{i} \in \mathcal{B}_{1}$$, is dense in $$L^{1}(\mathbf{R}^{n},\mathcal{B}_{1})$$. Indeed, by Proposition 5.5.6 (b) it suffices to approximate a $$\mathcal{C}_{0}^{\infty }\otimes \mathcal{B}_{1}$$-valued function with a $$\mathcal{Q}\otimes \mathcal{B}_{1}$$-valued function. But this is immediate since any function in $$\mathcal{C}_{0}^{\infty }(\mathbf{R}^{n})$$ can be approximated in $$L^{1}(\mathbf{R}^{n})$$ by finite linear combinations of characteristic functions of disjoint dyadic cubes.

Case 1: r = ∞. We fix $$F =\sum _{ i=1}^{m}\chi _{R_{i}}u_{i}$$ in $$\mathcal{Q}\otimes \mathcal{B}_{1}$$ and we notice that for each x ∈ R n we have $$\|F(x)\|_{\mathcal{B}_{1}} =\sum _{ i=1}^{m}\chi _{R_{i}}(x)\|u_{i}\|_{\mathcal{B}_{1}}$$, which is also a finite linear combination of characteristic functions of dyadic cubes. Apply the Calderón-Zygmund decomposition to $$\|F\|_{\mathcal{B}_{1}}$$ at height γ α, where $$\gamma = 2^{-n-1}B_{\star }^{-1}$$ as in the proof of Theorem 6.​3.​1.​ We extract a finite collection of closed dyadic cubes {Q j } j satisfying $$\sum _{j}\vert Q_{j}\vert \leq (\gamma \alpha )^{-1}\|F\|_{L^{1}(\mathbf{R}^{n},\mathcal{B}_{1})}$$ and we define the good function of the decomposition

Also define the bad function B(x) = F(x) − G(x). Then $$B(x) =\sum _{j}B_{j}(x)$$, where each B j is supported in Q j and has mean value zero over Q j . Moreover,

(5.6.8)

(5.6.9)

and $$\|B_{j}\|_{L^{1}(\mathbf{R}^{n},\mathcal{B}_{1})} \leq 2^{n+1}\gamma \alpha \vert Q_{j}\vert $$, by an argument similar to that given in the proof of Theorem 5.3.1. We only verify (5.6.9). On the cube Q j , G is equal to the constant $$\vert Q_{j}\vert ^{-1}\int _{Q_{j}}F(x)\,dx$$, and this is bounded by 2 n γ α. For each $$x \in \mathbf{R}^{n}\setminus \bigcup _{j}Q_{j}$$ and for each $$k = 0,1,2,\ldots$$ there exists a unique nonselected dyadic cube Q x (k) of generation k that contains x. Then for each k ≥ 0, we have

The intersection of the closures of the cubes Q x (k) is the singleton {x}. Using Corollary 2.​1.​16, we deduce that for almost all $$x \in \mathbf{R}^{n}\setminus \bigcup _{j}Q_{j}$$ we have

Since these averages are at most γ α, we conclude that $$\|F\|_{\mathcal{B}_{1}} \leq \gamma \alpha$$ almost everywhere on $$\mathbf{R}^{n}\setminus \bigcup _{j}Q_{j}$$, hence $$\|G\|_{\mathcal{B}_{1}} \leq \gamma \alpha$$ a.e. on this set. This proves (5.6.9).

By assumption we have

Then the set $$\big\{x \in \mathbf{R}^{n}:\,\,\|\mathbf{ T}(G)(x)\|_{\mathcal{B}_{2}} > \alpha /2\big\}$$ is null and we have

Let $$Q_{j}^{{\ast}} = 2\sqrt{n}\,Q_{j}.$$ We have

since $$B =\sum _{j}B_{j}.$$ It suffices to estimate the last sum. Denoting by y j is the center of the cube Q j and using the fact that B j has mean value zero over Q j , we write

where we used the fact that $$\vert x - y_{j}\vert \geq 2\vert y - y_{j}\vert $$ for all $$x\notin Q_{j}^{{\ast}}$$ and $$y \in Q_{j}$$ and (5.6.2). Consequently,

where $$C_{n}' = (2\sqrt{n})^{n}2^{n+1} + 2^{n+1}.$$ Thus $$\mathbf{T}$$ has an extension that maps $$L^{1}(\mathbf{R}^{n},\mathcal{B}_{1})$$ to $$L^{1,\infty }(\mathbf{R}^{n},\mathcal{B}_{2})$$ with constant C n (A + B ⋆). By interpolation (Exercise 5.5.3 (b)) it has an extension that satisfies (5.6.7).

Case 2: 1 < r < ∞. We fix $$F =\sum _{ i=1}^{m}\chi _{R_{i}}u_{i}$$ in $$\mathcal{Q}\otimes \mathcal{B}_{1}$$ and we notice that for each x ∈ R n we have $$\|F(x)\|_{\mathcal{B}_{1}} =\sum _{ i=1}^{m}\chi _{R_{i}}(x)\|u_{i}\|_{\mathcal{B}_{1}}$$. Thus the function $$x\mapsto \|F(x)\|_{\mathcal{B}_{1}}$$ is a finite linear combination of characteristic functions of disjoint dyadic cubes. We prove the weak type estimate (5.6.6) by applying the Calderón–Zygmund decomposition to the function $$x\mapsto \|F(x)\|_{\mathcal{B}_{1}}$$ defined on R n . Then we decompose F = G + B, where G and B satisfy properties analogous to the case r = ∞. The new ingredient in this case is that the set $$\big\{x \in \mathbf{R}^{n}:\,\,\|\mathbf{ T}(G)(x)\|_{\mathcal{B}_{2}} > \alpha /2\big\}$$ is not null but its measure can be estimated as follows:

where the first inequality is a consequence of the boundedness of $$\mathbf{T}$$ on L r and the second is obtained by combining (5.6.8) and (5.6.9). Combining this estimate for the good function with the one for the bad function obtained in the preceding case, it follows that $$\mathbf{T}$$ has an extension that satisfies (5.6.6), i.e., it maps $$\mathbf{T}:\,\, L^{1}(\mathbf{R}^{n},\mathcal{B}_{1})$$ to $$L^{1,\infty }(\mathbf{R}^{n},\mathcal{B}_{2})$$ with constant $$C_{n}'\,(A + B_{\star })$$, where $$C_{n}' = 2 + (2\sqrt{n})^{n}2^{n+1} + 2^{n+1}$$.

Next we interpolate between the estimates $$\mathbf{T}:\,\, L^{1}(\mathbf{R}^{n},\mathcal{B}_{1}) \rightarrow L^{1,\infty }(\mathbf{R}^{n},\mathcal{B}_{2})$$ and $$\mathbf{T}:\,\, L^{r}(\mathbf{R}^{n},\mathcal{B}_{1}) \rightarrow L^{r}(\mathbf{R}^{n},\mathcal{B}_{2})$$. Using Exercise 5.5.3 (b) and the fact that $$(p - 1)^{-1/p} \leq (p - 1)^{-1}$$ when 1 < p < 2, we obtain

(5.6.10)

when $$1 < p <\min (r,2)$$, where C n is independent of r, p, $$\mathcal{B}_{1}$$, and $$\mathcal{B}_{2}$$.

We prove (5.6.7) for p > r via duality. Since $$\mathbf{K}(x)$$ is an operator from $$\mathcal{B}_{1}$$ to $$\mathcal{B}_{2}$$, its adjoint $$\mathbf{K}^{{\ast}}(x)$$ is an operator from $$\mathcal{B}_{2}^{{\ast}}$$ to $$\mathcal{B}_{1}^{{\ast}}$$. Obviously $$\mathbf{K}^{{\ast}}(x)$$ and $$\mathbf{K}(x)$$ have the same norm, so (5.6.1) holds. For the same reason, condition (5.6.2) also holds for $$\mathbf{K}^{{\ast}}$$, Finally condition (5.6.3) also holds since for any $$\varepsilon _{k} \downarrow $$ as k → ∞ we have

Let $$\mathbf{T}^{{\prime}}$$ be the Banach-valued operator with kernel $$\mathbf{K}^{{\ast}}(-x)$$. Clearly $$\mathbf{T}^{{\prime}}$$ is well defined on $$\mathcal{C}_{0}^{\infty }\otimes \mathcal{B}_{2}^{{\ast}}$$. For $$F(y) =\sum _{ i=1}^{m}f_{i}(y)w_{i}^{{\ast}}$$ in $$\mathcal{C}_{0}^{\infty }\otimes \mathcal{B}_{2}^{{\ast}}$$ and $$G(z) =\sum _{ j=1}^{l}g_{j}(z)v_{j}$$ in $$\mathcal{C}_{0}^{\infty }\otimes \mathcal{B}_{1}$$ we prove the following duality relation

(5.6.11)

Indeed, for each index $$i \in \{ 1,\ldots,m\}$$ and $$j \in \{ 1,\ldots,l\}$$ we have

proving (5.6.11), provided we can justify the interchange the x (or z)-integral with the y-integral paired with the limit. These justifications can be given using the definition in (5.6.4). For the part of the y-integral where | y | ≥ 1 the interchange is easily justified in view of the absolute convergence of the double integral and (5.5.21). For the part of the y-integral where | y | ≤ 1 we introduce the operators $${\boldsymbol K_{0}}^{\!\!{\ast}}$$ and $$\mathbf{K}_{0}$$ and we use the facts that $$\vert g_{j}(z - y) - g_{j}(z)\vert \leq \|\nabla g_{j}\|_{L^{\infty }}\vert y\vert $$ and $$\vert f_{i}(x - y) - f_{i}(x)\vert \leq \|\nabla f_{i}\|_{L^{\infty }}\vert y\vert $$ together with the assumption $$\|\mathbf{K}(y)\|_{\mathcal{B}_{1}\rightarrow \mathcal{B}_{2}} \leq A\vert y\vert ^{-n}$$ and (5.5.21) to obtain the absolute convergence of the double integral, and thus justify the interchange.

We claim that $$\mathbf{T}^{{\prime}}$$ is bounded from $$L^{r'}(\mathbf{R}^{n},\mathcal{B}_{2}^{{\ast}})$$ to $$L^{r'}(\mathbf{R}^{n},\mathcal{B}_{1}^{{\ast}})$$. Indeed, to verify this assertion, we fix F in $$\mathcal{C}_{0}^{\infty }\otimes \mathcal{B}_{2}^{{\ast}}$$ and use Proposition 5.5.7 (b). Using (5.6.11), for each $$G \in \mathcal{C}_{0}^{\infty }\otimes \mathcal{B}_{1}$$, we write

so taking the supremum over all $$G \in \mathcal{C}_{0}^{\infty }\otimes \mathcal{B}_{1}$$ with $$\|G\|_{L^{r}(\mathbf{R}^{n},\,\mathcal{B}_{1})} \leq 1$$ we deduce that

Collecting these facts, we have that $$\mathbf{T}^{{\prime}}$$ is associated with a kernel $$\mathbf{K}^{{\ast}}(-x)$$ which satisfies (5.6.1), (5.6.2), and (5.6.3) (with $${\boldsymbol K_{0}}^{\!\!{\ast}}$$ in place of $$\mathbf{K}_{0}$$), and moreover it has a bounded extension that maps $$L^{r'}(\mathbf{R}^{n},\mathcal{B}_{2}^{{\ast}})$$ to $$L^{r'}(\mathbf{R}^{n},\mathcal{B}_{1}^{{\ast}})$$. The Calderón-Zygmund decomposition in the vector-valued setting (discussed in the first paragraph of the proof) yields that $$\mathbf{T}^{{\prime}}$$ has an extension that satisfies

Using interpolation (Exercise 5.5.3 (b)) and the fact that $$(p' - 1)^{1/p'} \leq p$$, we obtain that for 1 < p′ < r′, $$\mathbf{T}^{{\prime}}$$ has an extension on $$L^{p'}(\mathbf{R}^{n}\!,\mathcal{B}_{2}^{{\ast}})$$ that satisfies

(5.6.12)

Let $$F =\sum _{ i=1}^{m}\varphi _{i}\,u_{i}$$ be in the dense subspace $$\mathcal{C}_{0}^{\infty }\otimes \mathcal{B}_{1}$$ of $$L^{p}(\mathbf{R}^{n},\mathcal{B}_{1})$$. We observe that $$\|\mathbf{T}(F)\|_{L^{p}(\mathbf{R}^{n},\mathcal{B}_{2})} < \infty $$. Indeed, all $$\varphi _{i}$$ are supported in | x | ≤ R, then for | x | ≥ 2R we have

(5.6.13)

which is integrable to the power p > 1 in the region | x | ≥ 2R. Also using the definition in (5.6.4) we see that the expression on the left in (5.6.13) is bounded, hence integrable to the power p in the region $$\vert x\vert \leq 2R$$. For a fixed r < p < ∞, we are now able to apply Proposition 5.5.7 (a) to write

where we used (5.6.12). This combined with (5.6.10) implies the required conclusion whenever r < ∞ and $$p \in \big (1,\min (r,2)\big) \cup (r,\infty )$$. The remaining p’s follow by interpolation (Exercise 5.5.3 (a)). □ 

5.6.2 Applications

We proceed with some applications. An important consequence of Theorem 5.6.1 is the following:

Corollary 5.6.2.

Let A,B > 0 and let W j be a sequence of tempered distributions on R n whose Fourier transforms are uniformly bounded functions (i.e.,$$\vert \widehat{W_{j}}\vert \leq B$$ ). Suppose that for each j, W j coincides with a function K j on $$\mathbf{R}^{n}\setminus \{0\}$$ that satisfies

(5.6.14)

(5.6.15)

for some complex constant L j , and

(5.6.16)

Then there are constants C n ,C n ′ > 0 such that for all 1 < p,r < ∞ we have

where $$c(p,r) =\max (p,(p - 1)^{-1})\max (r,(r - 1)^{-1})$$ .

Proof.

Let T j be the operator given by convolution with the distribution W j . Clearly T j is L ² bounded with norm at most B. It follows from Theorem 5.3.3 that the T j ’s are of weak type (1, 1) and also bounded on L r with bounds at most a dimensional constant multiple of $$\max (r,(r - 1)^{-1})(A + B)$$, uniformly in j. We set $$\mathcal{B}_{1} = \mathcal{B}_{2} =\ell ^{r}$$ and define

for $$\{f_{j}\}_{j} \in L^{r}(\mathbf{R}^{n},\ell^{r})$$. It is immediate to verify that $$\mathbf{T}$$ maps $$L^{r}(\mathbf{R}^{n},\ell^{r})$$ to itself with norm at most a dimensional constant multiple of $$\max (r,(r - 1)^{-1})(A + B)$$. The kernel of $$\mathbf{T}$$ is $$\mathbf{K}$$ in L(ℓ r , ℓ r ) defined by

Obviously, we have

and therefore condition (5.6.3) holds for $$\mathbf{K}$$ as a consequence of (5.6.16). Moreover, (5.6.1) and (5.6.2) with $$\mathbf{K}_{0} =\{ L_{j}\}_{j}$$ are also valid for this $$\mathbf{K}$$, in view of assumptions (5.6.14) and (5.6.15). The desired conclusion follows from Theorem 5.6.1. □ 

If all the W j ’s are equal, we obtain the following corollary, which contains in particular the inequality (5.5.16) mentioned earlier.

Corollary 5.6.3.

Let W be an element of $$\mathcal{S}'(\mathbf{R}^{n})$$ whose Fourier transform is a function bounded in absolute value by some B > 0. Suppose that W coincides with some locally integrable function K on R n ∖{0} that satisfies

and

(5.6.17)

Let T be the operator given by convolution with W. Then there exist constants C n ,C n ′ > 0 such that for all 1 < p,r < ∞ we have that

where $$c(p,r) =\max (p,(p - 1)^{-1})\max (r,(r - 1)^{-1})$$ . In particular, these inequalities are valid for the Hilbert transform and the Riesz transforms.

Interestingly enough, we can use the very statement of Theorem 5.6.1 to obtain its corresponding vector-valued version.

Proposition 5.6.4.

Let let 1 < p,r < ∞ and let $$\mathcal{B}_{1}$$ and $$\mathcal{B}_{2}$$ be two Banach spaces. Suppose that $$\mathbf{T}$$ given by (5.6.4) is a bounded linear operator from $$L^{r}(\mathbf{R}^{n},\mathcal{B}_{1})$$ to $$L^{r}(\mathbf{R}^{n},\mathcal{B}_{2})$$ with norm B = B(r). Also assume that for all x ∈ R n ∖{0},$$\mathbf{K}(x)$$ is a bounded linear operator from $$\mathcal{B}_{1}$$ to $$\mathcal{B}_{2}$$ that satisfies conditions (5.6.1) , (5.6.2) , (5.6.3) for some A > 0 and $$\mathbf{K}_{0} \in L(\mathcal{B}_{1},\mathcal{B}_{2})$$ . Then there exist positive constants C n ,C n ′ such that for all $$\mathcal{B}_{1}$$ -valued functions F j we have

where $$c(p) =\max (p,(p\! -\! 1)^{-1})$$ .

Proof.

Let us denote by $$\ell^{r}(\mathcal{B}_{1})$$ the Banach space of all $$\mathcal{B}_{1}$$-valued sequences {u j } j that satisfy

Now consider the operator $$\mathbf{S}$$ defined on $$L^{r}(\mathbf{R}^{n},\ell^{r}(\mathcal{B}_{1}))$$ by

It is obvious that $$\mathbf{S}$$ maps $$L^{r}(\mathbf{R}^{n},\ell^{r}(\mathcal{B}_{1}))$$ to $$L^{r}(\mathbf{R}^{n},\ell^{r}(\mathcal{B}_{2}))$$ with norm at most B. Moreover, $$\mathbf{S}$$ has kernel $$\widetilde{K}(x) \in L(\ell^{r}(\mathcal{B}_{1}),\ell^{r}(\mathcal{B}_{2}))$$ given by

where $$\mathbf{K}$$ is the kernel of $$\mathbf{T}$$. It is not hard to verify that for x ∈ R n ∖{0} we have

hence for $$x\neq y \in \mathbf{R}^{n}$$ we also have

Moreover, if we define $$\widetilde{K_{0}} \in L\big(\ell^{r}(\mathcal{B}_{1}),\ell^{r}(\mathcal{B}_{2})\big)$$ by

for $$\{u_{j}\}_{j} \in \ell^{r}(\mathcal{B}_{1})$$, then we have

in $$L\big(\ell^{r}(\mathcal{B}_{1}),\ell^{r}(\mathcal{B}_{2})\big)$$.

We conclude that $$\widetilde{K}$$ satisfies conditions (5.6.1), (5.6.2), (5.6.3). Hence the operator $$\mathbf{S}$$ associate with $$\widetilde{K}$$ satisfies the conclusion of Theorem 5.6.1, that is, the desired inequalities for $$\mathbf{T}$$. □ 

5.6.3 Vector-Valued Estimates for Maximal Functions

Next, we discuss applications of vector-valued inequalities to some nonlinear operators. We fix an integrable function Φ on R n and for t > 0 define $$\varPhi _{t}(x) = t^{-n}\varPhi (t^{-1}x)$$. We suppose that Φ satisfies the following regularity condition:

(5.6.18)

We consider the maximal operator

defined for f in L ¹ + L ∞ . We are interested in obtaining L p estimates for M Φ . We observe that the trivial estimate

(5.6.19)

holds when p = ∞. It is natural to set

and view M Φ as the linear operator $$f\mapsto \{f {\ast}\varPhi _{\delta }\}_{\delta >0}$$ that maps $$\mathcal{B}_{1}$$-valued functions to $$\mathcal{B}_{2}$$-valued functions.

To do this precisely, for each x ∈ R n we define a bounded linear operator $$\mathbf{K}_{\varPhi }(x)$$ from $$\mathcal{B}_{1}$$ to $$\mathcal{B}_{2}$$ by setting for $$c \in \mathbf{C}$$

Clearly we have

Now (5.6.18) implies condition (5.6.2) for the kernel $$\mathbf{K}_{\varPhi }$$. Also condition (5.6.1) holds (for some A < depending on n) since

and also condition (5.6.3) holds since for every δ > 0 we have

We also define a $$\mathcal{B}_{2}$$-valued linear operator acting on complex-valued functions on R n by

Obvisouly $$\mathbf{M}_{\varPhi }$$ maps $$L^{\infty }(\mathbf{R}^{n},\mathcal{B}_{1})$$ to $$L^{\infty }(\mathbf{R}^{n},\mathcal{B}_{2})$$ with norm at most $$\|\varPhi \|_{L^{1}}$$.

Applying Theorem 5.6.1 with r = ∞ we obtain for 1 < p < ∞,

(5.6.20)

which can be immediately improved to

(5.6.21)

via interpolation with estimate (5.6.19) for all 1 < r < ∞.

Next we use estimate (5.6.21) to obtain vector-valued estimates for the sublinear operator M Φ .

Corollary 5.6.5.

Let Φ be an integrable function on R n that satisfies (5.6.18) . Then there exist dimensional constants C n and C n ′ such that for all 1 < p,r < ∞ the following vector-valued inequalities are valid:

(5.6.22)

where $$c(r) =\ 1 + (r\! -\! 1)^{-1}$$ , and

(5.6.23)

where $$c(p,r) =\big (1 + (r\! -\! 1)^{-1}\big)\big(p + (p - 1)^{-1}\big)$$ .

Proof.

We set $$\mathcal{B}_{1} = \mathbf{C}$$ and $$\mathcal{B}_{2} = L^{\infty }(\mathbf{R}^{+})$$. We use estimate (5.6.21) as a starting point in Proposition 5.6.4, which immediately yields the required conclusions (5.6.22) and (5.6.23). □ 

Similar estimates hold for the Hardy–Littlewood maximal operator.

Theorem 5.6.6.

For 1 < p,r < ∞ the Hardy–Littlewood maximal function M satisfies the vector-valued inequalities

(5.6.24)

(5.6.25)

where $$c(p,r) =\big (1 + (r\! -\! 1)^{-1}\big)\big(p + (p - 1)^{-1}\big)$$ .

Proof.

Let us fix a positive radial symmetrically decreasing Schwartz function Φ on R n that satisfies Φ(x) ≥ 1 when | x | ≤ 1. Then the Hardy–Littlewood maximal function M(f) is pointwise controlled by a constant multiple of the function M Φ ( | f | ). In view of Corollary 5.6.5, it suffices to check that for such a Φ, (5.6.18) holds. First observe that in view of the decreasing character of Φ, we have

and for this reason we choose to work with the easier dyadic maximal operator

We observe the validity of the simple inequalties

(5.6.26)

If we can show that

(5.6.27)

then (5.6.22) and (5.6.23) are satisfied with M Φ d replacing M Φ . We therefore turn our attention to (5.6.27). We have

where C N  > 0 depends on N > n, θ ∈ [0, 1], and | x −θ y | ≥ | x | ∕2 when | x | ≥ 2 | y | .

Now apply (5.6.22) and (5.6.23) to M Φ d and use (5.6.26) to obtain the desired vector-valued inequalities. □ 

Remark 5.6.7.

Observe that (5.6.24) and (5.6.25) also hold for r = ∞. These endpoint estimates can be proved directly by observing that

The same is true for estimates (5.6.22) and (5.6.23). Finally, estimates (5.6.25) and (5.6.23) also hold for p = ∞.

Exercises

5.6.1. (a) For all j ∈ Z, let I j be an interval in R and let T j be the operator given on the Fourier transform by multiplication by the characteristic function of I j . Prove that there exists a constant C > 0 such that for all 1 < p, r < ∞ and for all square integrable functions f j on R we have

(b) Let R j be arbitrary rectangles on R n with sides parallel to the axes and let S j be the operators given on the Fourier transform by multiplication by the characteristic functions of R j . Prove that there exists a dimensional constant C n  < ∞ such that for all indices 1 < p, r < ∞ and for all square integrable functions f j in $$L^{p}(\mathbf{R}^{n})$$ we have

[Hint: Part (a): Use Theorem 5.5.1 and the identity $$T_{j} = \frac{i} {2}\big(M^{a}HM^{-a} - M^{b}HM^{-b}\big)$$, if I j is χ (a, b), where $$M^{a}(f)(x) = f(x)e^{2\pi iax}$$ and H is the Hilbert transform. Part (b): Apply the result in part (a) in each variable.]

5.6.2. Let (T, d μ) be a σ-finite measure space. For every t ∈ T, let R(t) be a rectangle in R n with sides parallel to the axes such that the map $$t\mapsto R(t)$$ is measurable. Then there is a constant C n  > 0 such that for all 1 < p < ∞ and for all families of square integrable functions $$\{f_{t}\}_{t\in T}$$ on R n such that $$t\mapsto f_{t}(x)$$ is measurable for all x ∈ R n we have

[Hint: When n = 1 reduce matters to an $$L^{p}(L^{2}(T,d\mu ),L^{2}(T,d\mu ))$$ inequality for the Hilbert transform, via the hint in the preceding exercise. Verify the inequality p = 2 and then use Theorem 5.6.1 for the other p’s. Obtain the n-dimensional inequality by iterating the one-dimensional.]

5.6.3. Let Φ be a function on R n that satisfies $$\sup _{x\in \mathbf{R}^{n}}\vert x\vert ^{n}\vert \varPhi (x)\vert \leq A$$ and

for all R ≥ 1, where η is a continuous increasing function on [0, 2] that satisfies η(0) = 0 and $$\int _{0}^{2}\frac{\eta (t)} {t} \,dt < \infty \,.$$

(a) Prove that (5.6.27) holds.

(b) Show that if Φ lies in L ¹(R n ), then the maximal function $$f\mapsto \sup _{j\in \mathbf{Z}}\vert f {\ast}\varPhi _{2^{j}}\vert $$ maps $$L^{p}(\mathbf{R}^{n})$$ to itself for 1 < p ≤ ∞.

[Hint: Part (a): Modify the calculation in the proof of Theorem 5.6.6. Part (b): Use Theorem 5.6.1 with r = ∞.]

5.6.4. (a) On R, take $$f_{j} =\chi _{[2^{j-1},2^{j}]}$$ to prove that inequality (5.6.25) fails when p = ∞ and $$1 < r < \infty $$.

(b) Again on $$\mathbf{R}$$, take N > 2 and $$f_{j} =\chi _{[\frac{j-1} {N}, \frac{j} {N}]}$$ for $$j = 1,2,\ldots,N$$ to prove that (5.6.25) fails when 1 < p < ∞ and r = 1.

5.6.5. Let K be an integrable function on the real line and assume that the operator $$f\mapsto f {\ast} K$$ is bounded on $$L^{p}(\mathbf{R})$$ for some $$1 < p < \infty $$. Prove that the vector-valued inequality

may fail in general when q < 1.

[Hint: Take $$K =\chi _{[-1,1]}$$ and $$f_{j} =\chi _{[\frac{j-1} {N}, \frac{j} {N}]}$$ for 1 ≤ j ≤ N.]

5.6.6. Let $$\{Q_{j}\}_{j}$$ be a countable collection of cubes in R n with disjoint interiors. Let c j be the center of the cube Q j and d j its diameter. For $$\varepsilon > 0$$, define the Marcinkiewicz function associated with the family {Q j } j as follows:

Prove that for some constants $$C_{n,\varepsilon,p}$$ and $$C_{n,\varepsilon }$$ one has

and consequently $$\int _{\mathbf{R}^{n}}M_{\varepsilon }(x)\,dx \leq C_{n,\varepsilon }\sum _{j}\vert Q_{j}\vert.$$

[Hint: Verify that

and use Theorem 5.6.6.]

HISTORICAL NOTES

The L p boundedness of the conjugate function on the circle was announced in 1924 by Riesz [292], but its first proof appeared three years later in [294]. In view of the identification of the Hilbert transform with the conjugate function, the L p boundedness of the Hilbert transform is also attributed to M. Riesz. Riesz’s proof was first given for p = 2k, k ∈ Z ⁺, via an argument similar to that in the proof of Theorem 4.​1.​7.​ For p ≠ 2k this proof relied on interpolation and was completed with the simultaneous publication of Riesz’s article on interpolation of bilinear forms [293]. The weak type (1, 1) property of the Hilbert transform is due to Kolmogorov [197]. Additional proofs of the boundedness of the Hilbert transform have been obtained by Stein [350], Loomis [230], and Calderón [41]. The proof of Theorem 5.1.7, based on identity (5.1.23), is a refinement of a proof given by Cotlar [75].

The norm of the conjugate function on $$L^{p}(\mathbf{T}^{1})$$, and consequently that of the Hilbert transform on L p (R), was shown by Gohberg and Krupnik [129] to be $$\cot (\pi /2p)$$ when p is a power of 2. Duality gives that this norm is $$\tan (\pi /2p)$$ for 1 < p ≤ 2 whenever p′ is a power of 2. Pichorides [282] extended this result to all 1 < p < ∞ by refining Calderón’s proof of Riesz’s theorem. This result was also independently obtained by B. Cole (unpublished). The direct and simplified proof for the Hilbert transform given in Exercise 5.1.12 is in Grafakos [130]. The norm of the operators $$\frac{1} {2}(I \pm iH)$$ for real-valued functions was found to be $$\frac{1} {2}\big[\min (\cos (\pi /2p),\sin (\pi /2p))\big]^{-1}$$ by Verbitsky [366] and later independently by Essén [108]. The norm of the same operators for complex-valued functions was shown to be equal to $$[\sin (\pi /p)]^{-1}$$ by Hollenbeck and Verbitsky [156]. Exact formulas for the L p norm, 1 ≤ p < ∞ of the Hilbert transform acting on a characteristic function were obtained by Laeng [211]. The best constant in the weak type (1, 1) estimate for the Hilbert transform is equal to $$(1 + \frac{1} {3^{2}} + \frac{1} {5^{2}} + \cdots \,)(1 - \frac{1} {3^{2}} + \frac{1} {5^{2}} -\cdots \,)^{-1}$$ as shown by Davis [91] using Brownian motion; an alternative proof was later obtained by Baernstein [17]. Iwaniec and Martin [175] showed that the norms of the Riesz transforms on $$L^{p}(\mathbf{R}^{n})$$ coincide with that of the Hilbert transform on $$L^{p}(\mathbf{R})$$ for 1 < p < ∞.

Operators of the kind T Ω as well as the stopping-time decomposition of Theorem 5.3.1 were introduced by Calderón and Zygmund [46]. In the same article, Calderón and Zygmund used this decomposition to prove Theorem 5.3.3 for operators of the form T Ω when Ω satisfies a certain weak smoothness condition. The more general condition (5.3.12) first appeared in Hörmander’s article [159]. A more flexible condition sufficient to yield weak type (1, 1) bounds is contained in the article of Duong and McIntosh [104]. Theorems 5.2.10 and 5.2.11 are also due to Calderón and Zygmund [48]. The latter article contains the method of rotations. Algebras of operators of the form T Ω were studied in [49]. For more information on algebras of singular integrals see the article of Calderón [44]. Theorem 5.4.1 is due to Benedek, Calderón, and Panzone [22], while Example 5.4.2 is taken from Muckenhoupt [259]. Theorem 5.4.5 is due to Riviere [296]. A weaker version of this theorem, applicable for smoother singular integrals such as the maximal Hilbert transform, was obtained by Cotlar [75] (Theorem 5.3.4). Improvements of the main inequality in Theorem 5.3.4 for homogeneous singular integrals were obtained by Mateu and Verdera [245] and Mateu, Orobitg, and Verdera [244]. For a general overview of singular integrals and their applications, one may consult the expository article of Calderón [43].

Part (a) of Theorem 5.5.1 is due to Marcinkiewicz and Zygmund [242], although the case p = q was proved earlier by Paley [273] with a larger constant. The values of r for which a general linear operator of weak or strong type (p, q) admits bounded ℓ r extensions are described in Rubio de Francia and Torrea [304]. The $$L^{p}$$ and weak $$L^{p}$$ spaces in Theorem 5.5.1 can be replaced by general Banach lattices, as shown by Krivine [206] using Grothendieck’s inequality. Hilbert-space-valued estimates for singular integrals were obtained by Benedek, Calderón, and Panzone [22]. Other operator-valued singular integral operators were studied by Rubio de Francia, Ruiz, and Torrea [303]. Banach-valued singular integrals are studied in great detail in the book of García-Cuerva and Rubio de Francia [122], which provides an excellent presentation of the subject. The ℓ r -valued estimates (5.5.16) for the Hilbert transform were first obtained by Boas and Bochner [28]. The corresponding vector-valued estimates for the Hardy–Littlewood maximal function in Theorem 5.6.6 are due to Fefferman and Stein [115]. Conditions of the form (5.6.18) have been applied to several situations and can be traced in Zo [386].

The sharpness of the logarithmic condition (5.2.24) was indicated by Weiss and Zygmund [372], who constructed an example of an integrable function Ω with vanishing integral on S ¹ satisfying $$\int _{\mathbf{S}^{n-1}}\vert \varOmega (\theta )\vert \log ^{+}\vert \varOmega (\theta )\vert \big(\log (2 +\log (2 + \vert \varOmega (\theta )\vert ))\big)^{-\delta }\,d\theta = \infty $$ for all δ > 0 and of a continuous function in L p (R ²) for all 1 < p < ∞ such that $$\limsup _{\varepsilon \rightarrow 0}\vert T_{\varOmega }^{(\varepsilon )}(f)(x)\vert = \infty $$ for almost all x ∈ R ². The proofs of Theorems 5.2.10 and 5.2.11 can be modified to give that if Ω is in the Hardy space H ¹ of S n−1, then T Ω and $$T_{\varOmega }^{({\ast})}$$ map L p to L p for 1 < p < ∞. For T Ω this fact was proved by Connett [72] and independently by Ricci and Weiss [289]; for T Ω (∗) this was proved by Fan and Pan [110] and independently by Grafakos and Stefanov [139]. The latter authors [138] also obtained that the logarithmic condition $$\mathop{\mbox{ ess.sup}}_{\vert \xi \vert =1}\int _{\mathbf{S}^{n-1 }}\vert \varOmega (\theta )\vert (\log \frac{1} {\vert \xi \cdot \theta \vert }\vert )^{1+\alpha }\,d\theta < \infty $$, α > 0, implies L p boundedness for T Ω and T Ω (∗) for some p ≠ 2. See also Fan, Guo, and Pan [109] as well as Ryabogin and Rubin [308] for extensions. Examples of functions Ω for which T Ω maps L p to L p for a certain range of p’s but not for other ranges of p’s is given in Grafakos, Honzík, and Ryabogin [132]. A different example of this sort was provided later by Honzík [158]; the range of p’s for which boundedness holds are different for these examples. Honzík [157] also constructed a delicate example of an integrable function Ω with mean value zero over $$\mathbf{S}^{1}$$ such that T Ω is bounded on L ²(R) but T Ω (∗) is not.

The relatively weak condition $$\vert \varOmega \vert \log ^{+}\vert \varOmega \vert \in L^{1}(\mathbf{S}^{n-1})$$ also implies weak type (1, 1) boundedness for operators T Ω . This was obtained by Seeger [317] and later extended by Tao [355] to situations in which there is no Fourier transform structure. Earlier partial results are in Christ and Rubio de Francia [63] and in the simultaneous work of Hofmann [155], both inspired by the work of Christ [60]. Soria and Sjögren [324] showed that for arbitrary Ω in L ¹(S n−1), T Ω is weak type (1, 1) when restricted to radial functions. Examples due to Christ (published in [139]) indicate that even for bounded functions Ω on S n−1, T Ω may not map the endpoint Hardy space H ¹(R n ) to L ¹(R n ). However, Seeger and Tao [318] have showed that T Ω always maps the Hardy space H ¹(R n ) to the Lorentz space L 1, 2(R n ) when | Ω | (log⁺ | Ω | )² is integrable over S n−1. This result is sharp in the sense that for such Ω, T Ω may not map H ¹(R n ) to L 1, q (R n ) when q < 2 in general. If T Ω maps H ¹(R n ) to itself, Daly and Phillips [87] (in dimension n = 2) and Daly [86] (in dimensions n ≥ 3) showed that Ω must lie in the Hardy space H ¹(S n−1). There are also results concerning the singular maximal operator $$M_{\varOmega }(f)(x) =\sup _{r>0} \frac{1} {v_{n}r^{n}}\int _{\vert y\vert \leq r}\vert f(x - y)\vert \,\vert \varOmega (y)\vert \,dy$$, where Ω is an integrable function on S n−1 of not necessarily vanishing integral. Such operators were studied by Fefferman [116], Christ [60], and Hudson [162]. An excellent treatment of several kinds of singular integral operators with rough kernels is contained in the book of Lu, Ding, and Yan [234].

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Footnotes

1

this condition could be replaced by the assumption that K is integrable over any compact set that does not contain the origin.

© Springer Science+Business Media New York 2014

Loukas GrafakosClassical Fourier AnalysisGraduate Texts in Mathematics24910.1007/978-1-4939-1194-3_6

6. Littlewood–Paley Theory and Multipliers

Loukas Grafakos1 

(1)

Department of Mathematics, University of Missouri, Columbia, MO, USA

In this chapter we are concerned with orthogonality properties of the Fourier transform. This orthogonality is easily understood on L ², but at this point it is not clear how it manifests itself on other spaces. Square functions introduce a way to express and quantify orthogonality of the Fourier transform on L p and other function spaces. The introduction of square functions in this setting was pioneered by Littlewood and Paley, and the theory that subsequently developed is named after them. The extent to which Littlewood–Paley theory characterizes function spaces is remarkable.

Historically, Littlewood–Paley theory first appeared in the context of one-dimen- sional Fourier series and depended on complex function theory. With the development of real-variable methods, the whole theory became independent of complex methods and was extended to R n . This is the approach that we follow in this chapter. It turns out that the Littlewood–Paley theory is intimately related to the Calderón–Zygmund theory introduced in the previous chapter. This connection is deep and far-reaching, and its central feature is that one is able to derive the main results of one theory from the other.

The thrust and power of the Littlewood–Paley theory become apparent in some of the applications we discuss in this chapter. Such applications include the derivation of certain multiplier theorems, that is, theorems that yield sufficient conditions for bounded functions to be L p multipliers. As a consequence of Littlewood–Paley theory we also prove that the lacunary partial Fourier integrals $$\int _{\vert \xi \vert \leq 2^{N}}\widehat{f}(\xi )e^{2\pi ix\cdot \xi }\,d\xi$$ converge almost everywhere to an L p function f on R n .

6.1 Littlewood–Paley Theory

We begin by examining more closely what we mean by orthogonality of the Fourier transform. If the functions f j defined on R n have Fourier transforms $$\widehat{f_{j}}$$ supported in disjoint sets, then they are orthogonal in the sense that

(6.1.1)

Unfortunately, when 2 is replaced by some p ≠ 2 in (6.1.1), the previous quantities may not even be comparable, as we show in Examples 6.1.8 and 6.1.9. The Littlewood–Paley theorem provides a substitute inequality to (6.1.1) expressing the fact that certain orthogonality considerations are also valid in $$L^{p}(\mathbf{R}^{n})$$.

6.1.1 The Littlewood–Paley Theorem

The orthogonality we are searching for is best seen in the context of one-dimensional Fourier series (which was the setting in which Littlewood and Paley formulated their result). The primary observation is that the exponential $$e^{2\pi i2^{k}x }$$ oscillates half as much as $$e^{2\pi i2^{k+1}x }$$ and is therefore nearly constant in each period of the latter. This observation was instrumental in the proof of Theorem 3.​6.​4, which implied in particular that for all 1 < p < ∞ we have

(6.1.2)

In other words, we can calculate the L p norm of $$\sum _{k=1}^{N}a_{k}e^{2\pi i2^{k}x }$$ in almost a precise fashion to obtain (modulo multiplicative constants) the same answer as in the L ² case. Similar calculations are valid for more general blocks of exponentials in the dyadic range $$\{2^{k} + 1,\ldots,2^{k+1} - 1\}$$, since the exponentials in each such block behave independently from those in each previous block. In particular, the L p integrability of a function on T ¹ is not affected by the randomization of the sign of its Fourier coefficients in the previous dyadic blocks. This is the intuition behind the Littlewood–Paley theorem.

Motivated by this discussion, we introduce the Littlewood–Paley operators in the continuous setting.

Definition 6.1.1.

Let Ψ be an integrable function on R n and j ∈ Z. We define the Littlewood–Paley operator Δ j associated with Ψ by

where $$\varPsi _{2^{-j}}(x) = 2^{jn}\varPsi (2^{j}x)$$ for all x in R n . Thus we have $$\widehat{\varPsi _{2^{-j}}}(\xi ) =\widehat{\varPsi } (2^{-j}\xi )$$ for all ξ in R n . We note that whenever Ψ is a Schwartz function and f is a tempered distribution, the quantity Δ j (f) is a well defined function.

These operators depend on the choice of the function Ψ; in most applications we choose Ψ to be a smooth function with compactly supported Fourier transform. Observe that if $$\widehat{\varPsi }$$ is supported in some annulus 0 < c ₁ <  | ξ |  < c ₂ < ∞, then the Fourier transform of Δ j is supported in the annulus $$c_{1}2^{j} < \vert \xi \vert < c_{2}2^{j}$$; in other words, it is localized near the frequency | ξ | ≈ 2 j . Thus the purpose of Δ j is to isolate the part of frequency of a function concentrated near | ξ | ≈ 2 j .

The square function associated with the Littlewood–Paley operators Δ j is defined by

This quadratic expression captures the intrinsic orthogonality of the function f.

Theorem 6.1.2.

( Littlewood–Paley theorem ) Suppose that Ψ is an integrable $$\mathcal{C}^{1}$$ function on R n with mean value zero that satisfies

(6.1.3)

Then there exists a constant C n < ∞ such that for all 1 < p < ∞ and all f in L p ( R n ) we have

(6.1.4)

There also exists a C n ′ < ∞ such that for all f in L ¹ ( R n ) we have

(6.1.5)

Conversely, let Ψ be a Schwartz function such that either $$\widehat{\varPsi }(0) = 0$$ and

(6.1.6)

or $$\widehat{\varPsi }$$ is compactly supported away from the origin and

(6.1.7)

Then there is a constant C n,Ψ , such that for any $$f \in \mathcal{S}'(\mathbf{R}^{n})$$ with $$\big(\sum _{j\in \mathbf{Z}}\vert \varDelta _{j}(f)\vert ^{2}\big)^{\frac{1} {2} }$$ in L p ( R n ) for some 1 < p < ∞, there exists a unique polynomial Q such that the tempered distribution f − Q coincides with an L p function, and we have

(6.1.8)

Consequently, if g lies in L p ( R n ) for some 1 < p < ∞, then

Proof.

We first prove (6.1.4) when p = 2. Using Plancherel’s theorem, we see that (6.1.4) is a consequence of the inequality

(6.1.9)

for some C n  < ∞. Because of (6.1.3), Fourier inversion holds for Ψ. Furthermore, Ψ has mean value zero and we may write

(6.1.10)

from which we obtain the estimate

(6.1.11)

For ξ = (ξ ₁, …, ξ n ) ≠ 0, let j be such that | ξ j  | ≥ | ξ k  | for all k ∈ { 1, …, n}. Integrate by parts with respect to $$\partial _{j}$$ in (6.1.10) to obtain

from which we deduce the estimate

(6.1.12)

We now break the sum in (6.1.9) into the parts where 2−j  | ξ | ≤ 1 and 2−j  | ξ | ≥ 1 and use (6.1.11) and (6.1.12), respectively, to obtain (6.1.9). (See also Exercise 6.1.2.) This proves (6.1.4) when p = 2.

We now turn our attention to the case p ≠ 2 in (6.1.4). We view (6.1.4) and (6.1.5) as vector-valued inequalities in the spirit of Section 5.​5.​ Define an operator $$\mathbf{T}$$ acting on functions on R n as follows:

The inequalities (6.1.4) and (6.1.5) we wish to prove say simply that $$\mathbf{T}$$ is a bounded operator from $$L^{p}(\mathbf{R}^{n},\mathbf{C})$$ to $$L^{p}(\mathbf{R}^{n},\ell^{2})$$ and from $$L^{1}(\mathbf{R}^{n},\mathbf{C})$$ to $$L^{1,\infty }(\mathbf{R}^{n},\ell^{2})$$. We just proved that this statement is true when p = 2, and therefore the first hypothesis of Theorem 5.​6.​1 is satisfied. We observe that the operator $$\mathbf{T}$$ can be written in the form

where for each x ∈ R n , $$\mathbf{K}(x)$$ is a bounded linear operator from $$\mathbf{C}$$ to $$\ell^{2}$$ given by

(6.1.13)

We clearly have that $$\|\mathbf{K}(x)\|_{\mathbf{C}\rightarrow \ell^{2}} =\big (\sum _{j}\vert \varPsi _{2^{-j}}(x)\vert ^{2}\big)^{\frac{1} {2} }$$, and to be able to apply Theorem 5.​6.​1 we need to know that for some constant C n we have

(6.1.14)

(6.1.15)

(6.1.16)

Of these, (6.1.14) is easily obtained using (6.1.3), (6.1.15) i.e. trivial, and so we focus on (6.1.16). Since Ψ is a $$\mathcal{C}^{1}$$ function, for | x | ≥ 2 | y | we have

(6.1.17)

We also have that

(6.1.18)

Taking the geometric mean of (6.1.17) and (6.1.18), we obtain for any γ ∈ [0, 1]

(6.1.19)

Using this estimate, when | x | ≥ 2 | y | , we obtain

where we used (6.1.19) with γ = 1 in the first sum and (6.1.19) with γ = 1∕2 in the second sum. Using this bound, we easily deduce (6.1.16) by integrating over the region | x | ≥ 2 | y | . Finally, using Theorem 5.​6.​1 we conclude the proofs of (6.1.4) and (6.1.5), which establishes one direction of the theorem.

We now turn to the converse direction. Let Δ j ∗ be the adjoint operator of Δ j given by $$\widehat{\varDelta _{j}^{{\ast}}f} =\widehat{ f}\,\overline{\widehat{\varPsi _{2^{-j}}}}$$. Let f be in $$\mathcal{S}'(\mathbf{R}^{n})$$. Then the series $$\sum _{j\in \mathbf{Z}}\varDelta _{j}^{{\ast}}\varDelta _{j}(f)$$ converges in $$\mathcal{S}'(\mathbf{R}^{n})$$. To see this, it suffices to show that the sequence of partial sums $$u_{N} =\sum _{\vert j\vert <N}\varDelta _{j}^{{\ast}}\varDelta _{j}(f)$$ converges in $$\mathcal{S}'$$. This means that if we test this sequence against a Schwartz function g, then it is a Cauchy sequence and hence it converges as N → ∞. But an easy argument using duality and the Cauchy–Schwarz and Hölder’s inequalities shows that for M > N we have

and this can be made small by picking M > N ≥ N ₀(g). Since the sequence $$\langle u_{N},g\rangle$$ is Cauchy, it converges to some Λ(g). Now it remains to show that the map g↦Λ(g) is a tempered distribution. Obviously Λ(g) is a linear functional. Also,

and since $$\|g\|_{L^{p'}}$$ is controlled by a finite number of Schwartz seminorms of g, it follows that Λ is in $$\mathcal{S}'$$. The distribution Λ is the limit of the series $$\sum _{j}\varDelta _{j}^{{\ast}}\varDelta _{j}$$.

Under hypothesis (6.1.6), the Fourier transform of the tempered distribution $$f -\sum _{j\in \mathbf{Z}}\varDelta _{j}^{{\ast}}\varDelta _{j}(f)$$ is supported at the origin. This implies that there exists a polynomial Q such that $$f - Q =\sum _{j\in \mathbf{Z}}\varDelta _{j}^{{\ast}}\varDelta _{j}(f)$$. Now let g be a Schwartz function. We have

(6.1.20)

having used the definition of the adjoint (Section 2.​5.​2), the Cauchy–Schwarz inequality, Hölder’s inequality, and (6.1.4). Taking the supremum over all g in L p′ with norm at most one, we obtain that the tempered distribution f − Q is a bounded linear functional on L p′. By the Riesz representation theorem, f − Q coincides with an L p function whose norm satisfies the estimate

We now show uniqueness. If Q ₁ is another polynomial, with f − Q ₁ ∈ L p , then Q − Q ₁ must be an L p function; but the only polynomial that lies in L p is the zero polynomial. This completes the proof of the converse of the theorem under hypothesis (6.1.6).

To obtain the same conclusion under the hypothesis (6.1.7) we argue in a similar way but we leave the details as an exercise. (One may adapt the argument in the proof of Corollary 6.1.7 to this setting.) □ 

Remark 6.1.3.

We make some observations. If $$\widehat{\varPsi }$$ is real-valued, then the operators Δ j are self-adjoint. Indeed,

Moreover, if Ψ is a radial function, we see that the operators Δ j are self-transpose, that is, they satisfy

Assume now that Ψ is both radial and has a real-valued Fourier transform. Suppose also that Ψ satisfies (6.1.3) and that it has mean value zero. Then the inequality

(6.1.21)

is true for sequences of functions $$\{f_{j}\}_{j}$$. To see this we use duality. Let

Then

Inequality (6.1.4) says that the operator $$\mathbf{T}$$ maps $$L^{p}(\mathbf{R}^{n},\mathbf{C})$$ to L p (R n , ℓ ²), and its dual statement is that $$\mathbf{T}^{{\ast}}$$ maps L p′(R n , ℓ ²) to $$L^{p'}(\mathbf{R}^{n},\mathbf{C})$$. This is exactly the statement in (6.1.21) if p is replaced by p′. Since p is any number in (1, ∞), (6.1.21) is proved.

6.1.2 Vector-Valued Analogues

We now obtain a vector-valued extension of Theorem 6.1.2. We have the following.

Proposition 6.1.4.

Let Ψ be an integrable $$\mathcal{C}^{1}$$ function on R n with mean value zero that satisfies (6.1.3) and let Δ j be the Littlewood–Paley operator associated with Ψ. Then there exists a constant C n < ∞ such that for all 1 < p,r < ∞ and all sequences of L p functions f j we have

where $$\widetilde{C}_{p,r} =\max (p,(p - 1)^{-1})\max (r,(r - 1)^{-1})$$ . Moreover, for some C n ′ > 0 and all sequences of L ¹ functions f j we have

In particular,

(6.1.22)

Proof.

We introduce Banach spaces $$\mathcal{B}_{1} = \mathbf{C}$$ and $$\mathcal{B}_{2} =\ell ^{2}$$ and for f ∈ L p (R n ) define an operator

In the proof of Theorem 6.1.2 we showed that $$\mathbf{T}$$ has a kernel $$\mathbf{K}$$ that satisfies condition (6.1.16). Furthermore, $$\mathbf{T}$$ obviously maps $$L^{r}(\mathbf{R}^{n},\mathbf{C})$$ to L r (R n , ℓ r ). Applying Proposition 5.​6.​4, we obtain the first two statements of the proposition. Restricting to k = j yields (6.1.22). □ 

6.1.3 L p Estimates for Square Functions Associated with Dyadic Sums

Let us pick a Schwartz function Ψ whose Fourier transform is compactly supported in the annulus 2⁻¹ ≤ | ξ | ≤ 2² such that (6.1.6) is satisfied. (Clearly (6.1.6) has no chance of being satisfied if $$\widehat{\varPsi }$$ is supported only in the annulus 1 ≤ | ξ | ≤ 2.) The Littlewood–Paley operation f↦Δ j (f) represents the smoothly truncated frequency localization of a function f near the dyadic annulus | ξ | ≈ 2 j . Theorem 6.1.2 says that the square function formed by these localizations has L p norm comparable to that of the original function. In other words, this square function characterizes the L p norm of a function. This is the main feature of Littlewood–Paley theory.

One may ask whether Theorem 6.1.2 still holds if the Littlewood–Paley operators Δ j are replaced by their nonsmooth versions

(6.1.23)

This question has a surprising answer that already signals that there may be some fundamental differences between one-dimensional and higher-dimensional Fourier analysis. The square function formed by the operators in (6.1.23) can be used to characterize $$L^{p}(\mathbf{R})$$ in the same way Δ j did, but not $$L^{p}(\mathbf{R}^{n})$$ when n > 1 and p ≠ 2. The problem lies in the fact that the characteristic function of the unit disk is not an L p multiplier on R n when n ≥ 2 unless p = 2; see Section 5.​1 in [131]. The one-dimensional result we alluded to earlier is the following.

For $$j \in \mathbf{Z}$$ we introduce the one-dimensional operator

(6.1.24)

where

and Δ j # is a version of the operator Δ j in which the characteristic function of the set 2 j  ≤ | ξ |  < 2 j+1 replaces the function $$\widehat{\varPsi }(2^{-j}\xi )$$.

Theorem 6.1.5.

There exists a constant C ₁ such that for all 1 < p < ∞ and all f in $$L^{p}(\mathbf{R})$$ we have

(6.1.25)

Proof.

Pick a Schwartz function ψ on the line whose Fourier transform is supported in the set $$2^{-1} \leq \vert \xi \vert \leq 2^{2}$$ and is equal to 1 on the set 1 ≤ | ξ | ≤ 2. Let Δ j be the Littlewood–Paley operator associated with ψ. Observe that $$\varDelta _{j}\varDelta _{j}^{\#} =\varDelta _{ j}^{\#}\varDelta _{j} =\varDelta _{ j}^{\#}$$, since $$\widehat{\psi }$$ is equal to one on the support of $$\varDelta _{j}^{\#}(f)\,\widehat{\,}\,$$. We now use Exercise 5.6.1(a) to obtain

where the last inequality follows from Theorem 6.1.2. The reverse inequality for 1 < p < ∞ follows just like the reverse inequality (6.1.8) of Theorem 6.1.2 by simply replacing the Δ j ’s by the Δ j # ’s and setting the polynomial Q equal to zero. (There is no need to use the Riesz representation theorem here, just the fact that the L p norm of f can be realized as the supremum of expressions $$\vert \langle f,g\rangle \vert $$ where g has L p′ norm at most 1.) □ 

There is a higher-dimensional version of Theorem 6.1.5 with dyadic rectangles replacing the dyadic intervals. As has already been pointed out, the higher-dimensional version with dyadic annuli replacing the dyadic intervals is false.

Let us introduce some notation. For $$j \in \mathbf{Z}$$, we denote by I j the dyadic set $$[2^{j},2^{j+1})\bigcup (-2^{j+1},-2^{j}]$$ as in the statement of Theorem 6.1.5. For $$j_{1},\ldots,j_{n} \in \mathbf{Z}$$ define a dyadic rectangle

in R n . Actually $$R_{j_{1},\ldots,j_{n}}$$ is not a rectangle but a union of 2 n rectangles; with some abuse of language we still call it a rectangle. For notational convenience we write

Observe that for different $$\mathbf{j},\mathbf{j}' \in \mathbf{Z}^{n}$$ the rectangles R j and R j′ have disjoint interiors and that the union of all the R j ’s is equal to R n ∖{0}. In other words, the family of R j ’s, where j ∈ Z n , forms a tiling of R n , which we call the dyadic decomposition of R n . We now introduce operators

(6.1.26)

and we have the following n-dimensional extension of Theorem 6.1.5.

Theorem 6.1.6.

For a Schwartz function ψ on the line with integral zero we define the operator

(6.1.27)

where j = (j ₁ ,…,j n ) ∈ Z n . Then there is a dimensional constant C n such that

(6.1.28)

Let Δ j # be the operators defined in (6.1.26) . Then there exists a positive constant C n such that for all 1 < p < ∞ and all f ∈ L p ( R n ) we have

(6.1.29)

Proof.

We first prove (6.1.28). Note that if j = (j ₁, …, j n ) ∈ Z n , then the operator Δ j is equal to

where the $$\varDelta _{j_{r}}^{(j_{r})}$$ are one-dimensional operators given on the Fourier transform by multiplication by $$\widehat{\psi }(2^{-j_{r}}\xi _{r})$$, with the remaining variables fixed. Inequality in (6.1.28) is a consequence of the one-dimensional case. For instance, we discuss the case n = 2. Using Proposition 6.1.4, we obtain

where we also used Theorem 6.1.2 in the calculation. Higher-dimensional versions of this estimate may easily be obtained by induction.

We now turn to the upper inequality in (6.1.29). We pick a Schwartz function ψ whose Fourier transform is supported in the union $$[-4,-1/2]\bigcup [1/2,4]$$ and is equal to 1 on $$[-2,-1]\bigcup [1,2]$$. Then we clearly have

since $$\widehat{\psi }(2^{-j_{1}}\xi _{1})\cdots \widehat{\psi }(2^{-j_{n}}\xi _{n})$$ is equal to 1 on the rectangle R j . We now use Exercise 5.6.1(b) and estimate (6.1.28) to obtain

The lower inequality in (6.1.29) for 1 < p < ∞ is proved like inequality (6.1.8) in Theorem 6.1.2. The fundamental ingredient in the proof is that $$f =\sum _{\mathbf{j}\in \mathbf{Z}^{n}}\varDelta _{\mathbf{j}}^{\#}\varDelta _{\mathbf{j}}^{\#}(f)$$ for all Schwartz functions f, where the sum is interpreted as the L ²-limit of the sequence of partial sums. Thus the series converges in $$\mathcal{S}'$$, and pairing with a Schwartz function $$\overline{g}$$, we obtain the lower inequality in (6.1.29) for Schwartz functions, by applying the steps that prove (6.1.20) (with Q = 0). To prove the lower inequality in (6.1.29) for a general function $$f \in L^{p}(\mathbf{R}^{n})$$ we approximate an L p function by a sequence of Schwartz functions in the L p norm. Then both sides of the lower inequality in (6.1.29) for the approximating sequence converge to the corresponding sides of the lower inequality in (6.1.29) for f; the convergence of the sequence of L p norms of the square functions requires the upper inequality in (6.1.29) that was previously established. This concludes the proof of the theorem. □ 

Next we observe that if the Schwartz function ψ is suitably chosen, then the reverse inequality in estimate (6.1.28) also holds. More precisely, suppose $$\widehat{\psi }(\xi )$$ is an even smooth real-valued function supported in the set $$\frac{9} {10} \leq \vert \xi \vert \leq \frac{21} {10}$$ in $$\mathbf{R}$$ that satisfies

(6.1.30)

then we have the following.

Corollary 6.1.7.

Suppose that ψ satisfies (6.1.30) and let $$\varDelta _{\mathbf{j}}$$ be as in (6.1.27) . Let f be an L p function on R n such that the function $$\big(\sum _{\mathbf{j}\in \mathbf{Z}^{n}}\vert \varDelta _{\mathbf{j}}(f)\vert ^{2}\big)^{\!\frac{1} {2} }$$ is in $$L^{p}(\mathbf{R}^{n})$$ . Then there is a constant C n that depends only on the dimension and ψ such that the lower estimate

(6.1.31)

holds.

Proof.

If we had $$\sum _{j\in \mathbf{Z}}\vert \widehat{\psi }(2^{-j}\xi )\vert ^{2} = 1$$ instead of (6.1.30), then we could apply the method used in the lower estimate of Theorem 6.1.2 to obtain the required conclusion. In this case we provide another argument that is very similar in spirit.

We first prove (6.1.31) for Schwartz functions f. Then the series $$\sum _{j\in \mathbf{Z}^{n}}\varDelta _{\mathbf{j}}(f)$$ converges in L ² (and hence in $$\mathcal{S}'$$) to f. Now let g be another Schwartz function. We express the inner product $$\big\langle f,\overline{g}\,\big\rangle$$ as the action of the distribution $$\sum _{\mathbf{j\in \mathbf{Z}^{n}}}\varDelta _{\mathbf{j}}(f)$$ on the test function $$\overline{g}$$:

where we used the fact that Δ j (f) and Δ k (g) are orthogonal operators unless every coordinate of k is within 1 unit of the corresponding coordinate of j; this is an easy consequence of the support properties of $$\widehat{\psi }$$. We now take the supremum over all g in L p′ with norm at most 1, to obtain (6.1.31) for Schwartz functions f.

To extend this estimate to general L p functions f, we use the density argument described in the last paragraph in the proof of Theorem 6.1.6. □ 

6.1.4 Lack of Orthogonality on L p

We discuss two examples indicating why (6.1.1) cannot hold if the exponent 2 is replaced by some other exponent q ≠ 2. More precisely, we show that if the functions f j have Fourier transforms supported in disjoint sets, then the inequality

(6.1.32)

cannot hold if p > 2, and similarly, the inequality

(6.1.33)

cannot hold if p < 2. In both (6.1.32) and (6.1.33) the constants C p are supposed to be independent of the functions f j .

Example 6.1.8.

Pick a Schwartz function ζ whose Fourier transform is positive and supported in the interval | ξ | ≤ 1∕4. Let N be a large integer and let

.

Then

and the $$\widehat{f_{j}}$$’s have disjoint Fourier transforms. We obviously have

On the other hand, we have the estimate

since ζ does not vanish in a neighborhood of zero. We conclude that (6.1.32) cannot hold for this choice of f j ’s for p > 2.

Example 6.1.9.

We now indicate why (6.1.33) cannot hold for p < 2. We pick a smooth function Ψ on the line whose Fourier transform $$\widehat{\varPsi }$$ is supported in $$\big[\tfrac{7} {8}, \tfrac{17} {8} \big]$$, is nonnegative, is equal to 1 on $$\big[\tfrac{9} {8}, \tfrac{15} {8} \big]$$, and satisfies

Extend $$\widehat{\varPsi }$$ to be an even function on the whole line and let Δ j be the Littlewood–Paley operator associated with Ψ. Also pick a nonzero Schwartz function $$\varphi$$ on the real line whose Fourier transform is nonnegative and supported in the set $$\big[\tfrac{11} {8}, \tfrac{13} {8} \big]$$. Fix N a large positive integer and let

(6.1.34)

for j = 1, 2, …, N. Then the function $$\widehat{f_{j}}(\xi ) =\widehat{\varphi } (\xi -\frac{12} {8} 2^{j})$$ is supported in the set $$\big[\tfrac{11} {8} + \tfrac{12} {8} 2^{j}, \tfrac{13} {8} + \tfrac{12} {8} 2^{j}\big]$$, which is contained in $$\big[\tfrac{9} {8}2^{j}, \tfrac{15} {8} 2^{j}\big]$$ for j ≥ 3. In other words, $$\widehat{\varPsi }(2^{-j}\xi )$$ is equal to 1 on the support of $$\widehat{f_{j}}$$. This implies that

This observation combined with (6.1.21) gives for N ≥ 3,

where 1 < p < ∞. On the other hand, (6.1.34) trivially yields that

Letting N → ∞ we see that (6.1.33) cannot hold for p < 2 even when the f j ’s have Fourier transforms supported in disjoint sets.

Example 6.1.10.

A similar idea illustrates the necessity of the ℓ ² norm in (6.1.4). To see this, let Ψ and Δ j be as in Example 6.1.9. Let us fix 1 < p < ∞ and q < 2. We show that the inequality

(6.1.35)

cannot hold. Take $$f =\sum _{ j=3}^{N}f_{j}$$, where the f j are as in (6.1.34) and N ≥ 3. Then the left-hand side of (6.1.35) is bounded from below by $$\|\varphi \|_{L^{p}}(N - 2)^{1/q}$$, while the right-hand side is bounded above by $$\|\varphi \|_{L^{p}}(N - 2)^{1/2}$$. Letting N → ∞, we deduce that (6.1.35) is impossible when q < 2.

Example 6.1.11.

For 1 < p < ∞ and 2 < q < ∞, the inequality

(6.1.36)

cannot hold even under assumption (6.1.6) on Ψ. Let Δ j be as in Example 6.1.9. Let us suppose that (6.1.36) did hold for some q > 2 for these Δ j ’s. Then the self-adjointness of the Δ j ’s and duality would give

where the next-to-last inequality follows from (6.1.22) applied twice, while the one before that follows from support considerations. But since q′ < 2, this exactly proves (6.1.35), previously shown to be false, a contradiction.

We conclude that if both assertions (6.1.4) and (6.1.8) of Theorem 6.1.2 were to hold, then the ℓ ² norm inside the L p norm could not be replaced by an ℓ q norm for some q ≠ 2. Exercise 6.1.6 indicates the crucial use of the fact that ℓ ² is a Hilbert space in the converse inequality (6.1.8) of Theorem 6.1.2.

Exercises

6.1.1. Construct a Schwartz function Ψ that satisfies $$\sum _{j\in \mathbf{Z}}\vert \widehat{\varPsi }(2^{-j}\xi )\vert ^{2} = 1$$ for all ξ ∈ R n ∖{0} and whose Fourier transform is supported in the annulus $$\tfrac{6} {7} \leq \vert \xi \vert \leq 2$$ and is equal to 1 on the annulus $$1 \leq \vert \xi \vert \leq \tfrac{14} {7}$$.

[Hint: Set $$\widehat{\varPsi }(\xi ) =\eta (\xi )\big(\sum _{k\in \mathbf{Z}}\vert \eta (2^{-k}\xi )\vert ^{2}\big)^{-1/2}$$ for a suitable $$\eta \in \mathcal{C}_{0}^{\infty }(\mathbf{R}^{n})$$.]

6.1.2. Suppose that Ψ is an integrable function on R n that satisfies $$\vert \widehat{\varPsi }(\xi )\vert \leq B\,\min (\vert \xi \vert ^{\varepsilon },\vert \xi \vert ^{-\varepsilon '})$$ for some $$\varepsilon ',\varepsilon > 0$$. Show that for some constant $$C_{\varepsilon,\varepsilon '} < \infty $$ we have

6.1.3. Let Ψ be an integrable function on R n with mean value zero that satisfies

for some $$B,\varepsilon ',\varepsilon > 0$$ and for all y ≠ 0.

(a) Prove that $$\vert \widehat{\varPsi }(\xi )\vert \leq c_{n,\varepsilon,\varepsilon '}\,B\,\min (\vert \xi \vert ^{\min ( \frac{\varepsilon }{ 2},1)},\vert \xi \vert ^{-\varepsilon })$$ for some constant $$c_{n,\varepsilon,\varepsilon '}$$ and conclude that (6.1.4) holds for p = 2.

(b) Deduce the validity of (6.1.4) and (6.1.5).

(c) If $$\varepsilon < 1$$ and the assumption $$\vert \varPsi (x)\vert \leq B\,(1 + \vert x\vert )^{-n-\varepsilon }$$ is weakened to $$\vert \varPsi (x)\vert \leq B\,\vert x\vert ^{-n-\varepsilon }$$ for all x ∈ R n , then show that $$\vert \widehat{\varPsi }(\xi )\vert \leq c_{n,\varepsilon,\varepsilon '}\,B\,\min (\vert \xi \vert ^{\frac{\varepsilon }{ 2} },\vert \xi \vert ^{-\varepsilon })$$ and thus (6.1.4) and (6.1.5) are valid.

[Hint: Part (a): Make use of the identity

where $$y = \tfrac{1} {2} \tfrac{\xi } {\vert \xi \vert ^{2}}$$ when | ξ | ≥ 1. For | ξ | ≤ 1 use the mean value property of Ψ to write $$\widehat{\varPsi }(\xi ) =\int _{\vert x\vert \leq 1}\varPsi (x)(e^{-2\pi ix\cdot \xi }- 1)\,dx$$ and split the integral in the regions | x | ≤ 1 and | x | ≥ 1. Part (b): If $$\mathbf{K}$$ is defined by (6.1.13), then control the $$\ell^{2}(\mathbf{Z})$$ norm by the $$\ell^{1}(\mathbf{Z})$$ norm to prove (6.1.16). Then split the sum $$\sum _{j\in \mathbf{Z}}\int _{\vert x\vert \geq 2\vert y\vert }\big\vert \varPsi _{2^{-j}}(x - y) -\varPsi _{2^{-j}}(x)\big\vert \,dx$$ into the parts $$\sum _{2^{j}\leq \vert y\vert ^{-1}}$$ and $$\sum _{2^{j}>\vert y\vert ^{-1}}$$. Part (c): Notice that when $$\varepsilon < 1$$, we have $$\vert \int _{\vert x\vert \leq 1}\varPsi (x)(e^{-2\pi ix\cdot \xi }- 1)\,dx\vert \leq C_{n}\,B\,\vert \xi \vert ^{\frac{\varepsilon }{ 2} }$$. ]

6.1.4. Let Ψ be an integrable function on R n with mean value zero that satisfies

for some $$B,\varepsilon ',\varepsilon > 0$$ and for all y ≠ 0. Let $$\varPsi _{t}(x) = t^{-n}\varPsi (x/t)$$. (a) Prove that there are constants $$c_{n},c_{n}'$$ such that

(b) Show that there exist constants C n , C n ′ such that for all $$1 < p < \infty $$ and for all $$f \in L^{p}(\mathbf{R}^{n})$$ we have

and also for all f ∈ L ¹(R n ) we have

(c) Under the additional hypothesis that $$0 <\int _{ 0}^{\infty }\vert \widehat{\varPsi }(t\xi )\vert ^{2}\frac{dt} {t} = c_{0}$$ for all $$\xi \in \mathbf{R}^{n}\setminus \{0\}$$, prove that for all $$f \in L^{p}(\mathbf{R}^{n})$$ we have

[Hint: Part (a): Use the Cauchy-Schwarz inequality to obtain

and split the integral on the right into the regions t ≤ | y | and t >  | y | . In the second region use that Ψ is bounded to replace the square by the first power. Part (b): Use Exercise 6.1.2 and part (a) of Exercise 6.1.3 and to deduce the inequality when p = 2. Then apply Theorem 5.​6.​1.​ Part (c): Prove the inequality first for $$f \in \mathcal{S}(\mathbf{R}^{n})$$ using duality.]

6.1.5. Prove the following generalization of Theorem 6.1.2. Let A > 0. Suppose that $$\{K_{j}\}_{j\in \mathbf{Z}}$$ is a sequence of locally integrable functions on $$\mathbf{R}^{n}\setminus \{0\}$$ that satisfies

and for each $$j \in \mathbf{Z}$$ there is a number L j such that

If the K j coincide with tempered distributions W j that satisfy

then the operator

maps $$L^{p}(\mathbf{R}^{n})$$ to itself and is weak type (1, 1) norms at most multiples of A + B.

6.1.6.

Suppose that $$\mathcal{H}$$ is a Hilbert space with inner product $$\langle \,\cdot \,,\,\cdot \,\rangle _{\mathcal{H}}$$. Let A > 0 and 1 < p < ∞. Suppose that an operator T from $$L^{2}(\mathbf{R}^{n}) \rightarrow L^{2}(\mathbf{R}^{n},\mathcal{H})$$ is a multiple of an isometry, that is,

for all $$g \in L^{2}(\mathbf{R}^{n},\mathcal{H})$$. Then the inequality $$\|T(f)\|_{L^{p}(\mathbf{R}^{n},\mathcal{H})} \leq C_{p}\|f\|_{L^{p}(\mathbf{R}^{n})}$$ for all $$f \in \mathcal{S}(\mathbf{R}^{n})$$ implies

for all in $$f \in \mathcal{S}(\mathbf{R}^{n})$$.

[Hint: Use the inner product structure and polarization to obtain

and then argue as in the proof of inequality (6.1.8).]

6.1.7.

Suppose that $$\{m_{j}\}_{j\in \mathbf{Z}}$$ is a sequence of bounded functions supported in the intervals [2 j , 2 j+1]. Let $$T_{j}(f) = (\widehat{f}\,m_{j})\,^{\vee }\,$$ be the corresponding multiplier operators. Assume that for all sequences of functions $$\{f_{j}\}_{j}$$ the vector-valued inequality

is valid for some 1 < p < ∞. Prove there is a C p  > 0 such that for all finite subsets S of $$\mathbf{Z}$$ we have

[Hint: Use that $$\big\langle \sum _{j\in S}T_{j}(f),g\big\rangle =\sum _{j\in S}\big\langle \varDelta _{j}^{\#}T_{j}(f),\varDelta _{j}^{\#}(g)\big\rangle$$.]

6.1.8. Let m be a bounded function on R n that is supported in the annulus 1 ≤ | ξ | ≤ 2 and define $$T_{j}(f) =\big (\widehat{f}(\xi )m(2^{-j}\xi )\big)\,^{\vee }\,$$. Suppose that the square function $$f\mapsto \big(\sum _{j\in \mathbf{Z}}\vert T_{j}(f)\vert ^{2}\big)^{1/2}$$ is bounded on L p (R n ) for some 1 < p < ∞. Show that for every finite subset S of the integers we have

for some constant C p, n independent of S.

6.1.9. Fix a nonzero Schwartz function h on the line whose Fourier transform is supported in the interval $$\big[-\frac{1} {8}, \frac{1} {8}\big]$$. For {a j } a sequence of numbers, set

Prove that for all 1 < p < ∞ there exists a constant C p such that

[Hint: Write $$f =\sum _{ j=1}^{\infty }\varDelta _{j}(a_{j}e^{2\pi i2^{j}(\cdot ) }h)$$, where Δ j is given by convolution with $$\varphi _{2^{-j}}$$ for some φ whose Fourier transform is supported in the interval $$\big[\frac{6} {8}, \frac{10} {8} \big]$$ and is equal to 1 on $$\big[\frac{7} {8}, \frac{9} {8}\big]$$. Then use (6.1.21).]

6.1.10. Let Ψ be a Schwartz function whose Fourier transform is supported in the annulus $$\frac{1} {2} \leq \vert \xi \vert \leq 2$$ and that satisfies (6.1.7). Define a Schwartz function Φ by setting

Let S ₀ be the operator given by convolution with Φ. Let 1 < p < ∞ and f ∈ L p (R n ). Show that

[Hint: Use Theorem 6.1.2 together with the identity $$S_{0} +\sum _{ j=1}^{\infty }\varDelta _{j} = I$$.]

6.2 Two Multiplier Theorems

We now return to the spaces $$\mathcal{M}_{p}$$ introduced in Section 2.​5.​ We seek sufficient conditions on $$L^{\infty }$$ functions defined on $$\mathbf{R}^{n}$$ to be elements of $$\mathcal{M}_{p}$$. In this section we are concerned with two fundamental theorems that provide such sufficient conditions. These are the Marcinkiewicz and the Hörmander–Mihlin multiplier theorems. Both multiplier theorems are consequences of the Littlewood–Paley theory discussed in the previous section.

Using the dyadic decomposition of R n , we can write any L ∞ function m as the sum

where $$\mathbf{j} = (j_{1},\ldots,j_{n})$$, $$R_{\mathbf{j}} = I_{j_{1}} \times \ldots \times I_{j_{n}}$$, and $$I_{k} = [2^{k},2^{k+1})\bigcup (-2^{k+1},-2^{k}]$$. For j in Z n we set $$m_{\mathbf{j}} = m\chi _{R_{\mathbf{j}}}$$. A consequence of the ideas developed so far is the following characterization of $$\mathcal{M}_{p}(\mathbf{R}^{n})$$ in terms of a vector-valued inequality.

Proposition 6.2.1.

Let $$m \in L^{\infty }(\mathbf{R}^{n})$$ and let $$m_{\mathbf{j}} = m\chi _{R_{\mathbf{j}}}$$ . Then m lies in $$\mathcal{M}_{p}(\mathbf{R}^{n})$$ , that is, for some c p we have

if and only if for some C p > 0 we have

(6.2.1)

for all sequences of functions f j in $$L^{p}(\mathbf{R}^{n})$$ .

Proof.

Suppose that $$m \in \mathcal{M}_{p}(\mathbf{R}^{n})$$. Exercise 5.6.1 gives the first inequality below

while the second inequality follows from Theorem 5.​5.​1.​ (Observe that when p = q in Theorem 5.​5.​1, then C p, q  = 1.) Conversely, suppose that (6.2.1) holds for all sequences of functions f j . Fix a function f and apply (6.2.1) to the sequence $$(\widehat{f}\chi _{R_{\mathbf{j}}})\,^{\vee }\,$$, where R j is the dyadic rectangle indexed by j = (j ₁, …, j n ) ∈ Z n . We obtain

Using Theorem 6.1.6, we obtain that the previous inequality is equivalent to the inequality

which implies that $$m \in \mathcal{M}_{p}(\mathbf{R}^{n})$$. □ 

6.2.1 The Marcinkiewicz Multiplier Theorem on R

Proposition 6.2.1 suggests that the behavior of m on each dyadic rectangle R j should play a crucial role in determining whether m is an L p multiplier. The Marcinkiewicz multiplier theorem provides such sufficient conditions on m restricted to any dyadic rectangle R j . Before stating this theorem, we illustrate its main idea via the following example. Suppose that m is a bounded function that vanishes near −∞, that is differentiable at every point, and whose derivative is integrable. Then we may write

from which it follows that for a Schwartz function f we have

Since the operators $$f\mapsto (\widehat{f}\chi _{[t,\infty )})\,^{\vee }\,$$ map $$L^{p}(\mathbf{R})$$ to itself independently of t, it follows that

thus yielding that m is in $$\mathcal{M}_{p}(\mathbf{R})$$. The next multiplier theorem is an improvement of this result and is based on the Littlewood–Paley theorem. We begin with the one-dimensional case, which already captures the main ideas.

Theorem 6.2.2.

( Marcinkiewicz multiplier theorem ) Let $$m:\,\, \mathbf{R} \rightarrow \mathbf{R}$$ be a bounded function that is $$\mathcal{C}^{1}$$ in every dyadic set $$(2^{j},2^{j+1})\bigcup (-2^{j+1},-2^{j})$$ for $$j \in \mathbf{Z}$$ . Assume that the derivative m′ of m satisfies

(6.2.2)

Then for all 1 < p < ∞ we have that $$m \in \mathcal{M}_{p}(\mathbf{R})$$ and for some C > 0 we have

(6.2.3)

Proof.

Since the function m has an integrable derivative on (2 j , 2 j+1), it has bounded variation in this interval and hence it is a difference of two increasing functions. Therefore, m has left and right limits at the points 2 j and 2 j+1, and by redefining m at these points we may assume that m is right continuous at the points 2 j and left continuous at the points − 2 j .

Set $$I_{j} = [2^{j},2^{j+1}) \cup (-2^{j+1},-2^{j}]$$ and $$I_{j}^{+} = [2^{j},2^{j+1})$$ whenever $$j \in \mathbf{Z}$$. Given an interval I in $$\mathbf{R}$$, we introduce an operator Δ I defined by $$\varDelta _{I}(f) = (\widehat{f}\chi _{I})\,^{\vee }\,$$. With this notation $$\varDelta _{I_{j}^{+}}(f)$$ is “half” of the operator Δ j # introduced in the previous section. Given m as in the statement of the theorem, we write m(ξ) = m ₊(ξ) + m ₋(ξ), where m ₊(ξ) = m(ξ)χ ξ ≥ 0 and m ₋(ξ) = m(ξ)χ ξ < 0. We show that both m ₊ and m ₋ are L p multipliers. Since m′ is integrable over all intervals of the form [2 j , ξ] when 2 j  ≤ ξ < 2 j+1, the fundamental theorem of calculus gives

from which it follows that for a Schwartz function f on the real line we have

We therefore obtain the identity

which implies that

using the hypothesis (6.2.2). Taking ℓ ²(Z) norms we obtain

Exercise 5.6.2 gives

while the hypothesis on m′ implies the inequality

Using Theorem 6.1.5 we obtain that

and the latter is at most a constant multiple of $$\max (p,(p - 1)^{-1})^{3}\big\|f\big\|_{L^{p}}.$$ Putting things together we deduce that

(6.2.4)

from which we obtain the estimate

using the lower estimate of Theorem 6.1.5. This proves (6.2.3) for m ₊. A similar argument also works for m ₋, and this concludes the proof by summing the corresponding estimates for m ₊ and m ₋. □ 

We remark that the same proof applies under the more general assumption that m is a function of bounded variation on every interval [2 j , 2 j+1] and [−2 j+1, −2 j ]. In this case the measure | m′(t) | dt should be replaced by the total variation | dm(t) | of the Lebesgue–Stieltjes measure dm(t).

Example 6.2.3.

Any bounded function that is constant on dyadic intervals is an L p multiplier. Also, the function

is an L p multiplier on R for 1 < p < ∞.

6.2.2 The Marcinkiewicz Multiplier Theorem on R n

We now extend this theorem on R n . As usual we denote the coordinates of a point ξ ∈ R n by (ξ ₁, …, ξ n ). We recall the notation $$I_{j} = (-2^{j+1},-2^{j}]\bigcup [2^{j},2^{j+1})$$ and $$R_{\mathbf{j}} = I_{j_{1}} \times \ldots \times I_{j_{n}}$$ whenever $$\mathbf{j} = (j_{1},\ldots,j_{n}) \in \mathbf{Z}^{n}$$.

Theorem 6.2.4.

Let m be a bounded function on R n such that for all α = (α ₁ ,…,α n ) with |α ₁ |,…,|α n |≤ 1 the derivatives ∂ α m are continuous up to the boundary of R j for all j ∈ Z n . Assume that there is a constant A < ∞ such that for all partitions {s ₁ ,…,s k } ∪{ r ₁ ,…,r ℓ } ={ 1,2,…,n} with n = k + ℓ and all ξ ∈ R j we have

(6.2.5)

for all j = (j ₁ ,…,j n ) ∈ Z n . Then m is in $$\mathcal{M}_{p}(\mathbf{R}^{n})$$ whenever 1 < p < ∞ and there is a constant C n < ∞ such that

(6.2.6)

Proof.

We prove this theorem only in dimension n = 2, since the general case presents no substantial differences but only some notational inconvenience. We decompose the given function m as

where each of the last four terms is supported in one of the four quadrants. For instance, the function m ₊₋(ξ ₁, ξ ₂) is supported in the quadrant ξ ₁ ≥ 0 and ξ ₂ < 0. As in the one-dimensional case, we work with each of these pieces separately. By symmetry we choose to work with m ₊₊ in the following argument.

Using the fundamental theorem of calculus, we obtain the following simple identity, valid for $$2^{j_{1}} \leq \xi _{1} < 2^{j_{1}+1}$$ and $$2^{j_{2}} \leq \xi _{2} < 2^{j_{2}+1}$$:

(6.2.7)

We introduce operators $$\varDelta _{I}^{(r)}$$, r ∈ { 1, 2}, acting in the rth variable (with the other variable remaining fixed) given by multiplication on the Fourier transform side by the characteristic function of the interval I. Likewise, we introduce operators $$\varDelta _{j}^{\#(r)}$$, r ∈ { 1, 2} (also acting in the rth variable), given by multiplication on the Fourier transform side by the characteristic function of the set $$(-2^{j+1},-2^{j}]\bigcup [2^{j},2^{j+1})$$. For notational convenience, for a given Schwartz function f we write

and likewise we define f ₊₋, f ₋₊, and f ₋₋.

Multiplying both sides of (6.2.7) by the function $$\widehat{f}\chi _{R_{\mathbf{j}}}\chi _{(0,\infty )^{2}}$$ and taking inverse Fourier transforms yields

(6.2.8)

We apply the Cauchy–Schwarz inequality in the last three terms of (6.2.8) with respect to the measures $$\vert (\partial _{1}m)(t_{1},2^{j_{2}})\vert dt_{1}$$, $$\vert (\partial _{2}m)(2^{j_{1}},t_{2})\vert dt_{2}$$, $$\vert (\partial _{1}\partial _{2}m)(t_{1},t_{2})\vert \,dt_{2}dt_{1}$$ and we use hypothesis (6.2.5) to deduce

Both sides of the preceding inequality are sequences indexed by $$\mathbf{j} \in \mathbf{Z}^{2}$$. We apply $$\ell^{2}(\mathbf{Z}^{2})$$ norms and use Minkowski’s inequality to deduce the pointwise estimate

where ν is the counting measure $$\sum _{j\in \mathbf{Z}}\delta _{2^{j}}$$ defined by $$\nu (A) = \#\{j \in \mathbf{Z}:\,\, 2^{j} \in A\}$$ for subsets A of (0, ∞). We now take $$L^{p}(\mathbf{R}^{2})$$ norms and we estimate separately the contribution of each of the four terms on the right side. Using Exercise 5.6.2 we obtain

But the functions $$(t_{1},t_{2})\mapsto \varDelta _{[\log _{2}t_{1}]}^{\#(1)}\varDelta _{[\log _{2}t_{2}]}^{\#(2)}(f_{++})$$ are constant on products of intervals of the form $$[2^{j_{1}},2^{j_{1}+1}) \times [2^{j_{2}},2^{j_{2}+1})$$; hence using hypothesis (6.2.5) again we deduce the estimate

where the penultimate estimate follows from Theorem 6.1.6 and the last estimate by the boundedness of the Hilbert transform (Theorem 5.​1.​7). We now appeal to inequality (6.1.29) which yields the required estimate for the $$L^{p}(\mathbf{R}^{2})$$ norm of $$(\widehat{f}\,m_{++})\,^{\vee }\,$$. A similar argument also works for the remaining parts of m ₊₋, m ₋₊, m ₋₋, and summing concludes the proof of (6.2.6).

The analogous estimate on R n is

which is obtained in a similar fashion. Using (6.1.29), this implies that

A similar inequality holds when some (or all) +’s are replaced by −’s. □ 

We now give a condition that implies (6.2.5) and is well suited for a variety of applications.

Corollary 6.2.5.

Let m be a bounded $$\mathcal{C}^{n}$$ function defined away from the coordinate axes on R n . Assume that for all k ∈{ 1,…,n}, all distinct j ₁ ,…,j k ∈{ 1,2,…,n}, and all $$\xi _{r} \in \mathbf{R}\setminus \{0\}$$ for $$r\notin \{j_{1},\ldots,j_{k}\}$$ we have

(6.2.9)

Then m satisfies (6.2.6) .

Proof.

Simply observe that condition (6.2.9) implies (6.2.5). □ 

Example 6.2.6.

The following are examples of functions that satisfy the hypotheses of Corollary 6.2.5:

where α ₁ +α ₂ + … +α n  = α, α j  > 0,

The functions m ₁ and m ₂ are defined on R n ∖{0} and m ₃ on $$\mathbf{R}^{3}\setminus \{0\}$$.

The previous examples and many other examples that satisfy the hypothesis (6.2.9) of Corollary 6.2.5 are invariant under a set of dilations in the following sense: suppose that there exist $$k_{1},\ldots,k_{n} \in \mathbf{R}^{+}$$ and $$s \in \mathbf{R}$$ such that the smooth function m on R n ∖{0} satisfies

for all $$\xi _{1},\ldots,\xi _{n} \in \mathbf{R}$$ and λ > 0. Then m satisfies condition (6.2.9). Indeed, differentiation gives

for every multi-index α = (α ₁, …, α n ). Now for every ξ ∈ R n ∖{0} pick the unique λ ξ  > 0 such that $$(\lambda _{\xi }^{k_{1}}\xi _{1},\ldots,\lambda _{\xi }^{k_{n}}\xi _{n}) \in \mathbf{S}^{n-1}$$. Then $$\lambda _{\xi }^{k_{j}\alpha _{j}} \leq \vert \xi _{j}\vert ^{-\alpha _{j}}$$, and it follows that

6.2.3 The Mihlin–Hörmander Multiplier Theorem on R n

We now discuss another multiplier theorem that also requires decay of derivatives. We will consider the situation where each differentiation produces uniform decay in all variables, quantitatively expressed via the condition

(6.2.10)

for each multi-index α. The decay can also be expressed in terms of a square integrable estimate that has the form

(6.2.11)

for all multi-indices α and all R > 0. Obviously (6.2.10) implies (6.2.11)

Theorem 6.2.7.

Let m(ξ) be a complex-valued bounded function on R n ∖{0} that satisfies for some A < ∞

(6.2.12)

for all multi-indices |α|≤ [n∕2] + 1 and all R > 0.

Then for all 1 < p < ∞, m lies in $$\mathcal{M}_{p}(\mathbf{R}^{n})$$ and the following estimate is valid:

(6.2.13)

Moreover, the operator $$f\mapsto (\widehat{f}m)\,^{\vee }\,$$ maps L ¹ ( R n ) to L 1,∞ ( R n ) with norm at most a dimensional constant multiple of $$A +\| m\|_{L^{\infty }}$$ .

We remark that in most applications, condition (6.2.12) appears in the form

(6.2.14)

which should be, in principle, easier to verify.

Proof.

Since m is a bounded function, the operator given by convolution with $$W = m\,^{\vee }\,$$ is bounded on L ²(R n ). To prove that this operator maps L ¹(R n ) to L 1, ∞ (R n ), it suffices to prove that the distribution W coincides with a function K on R n ∖{0} that satisfies Hörmander’s condition.

Let $$\widehat{\zeta }$$ be a smooth function supported in the annulus $$\frac{1} {2} \leq \vert \xi \vert \leq 2$$ such that

Set $$m_{j}(\xi ) = m(\xi )\widehat{\zeta }(2^{-j}\xi )$$ for $$j \in \mathbf{Z}$$ and $$K_{j} = m_{j}\,^{\vee }\,$$. We begin by observing that $$\sum _{-N}^{N}K_{j}$$ converges to W in $$\mathcal{S}'(\mathbf{R}^{n})$$. Indeed, for all $$\varphi \in \mathcal{S}(\mathbf{R}^{n})$$ we have

We set $$n_{0} = [\tfrac{n} {2} ] + 1$$. We claim that there is a constant $$\widetilde{C}_{n}$$ such that

(6.2.15)

(6.2.16)

To prove (6.2.15) we multiply and divide the integrand in (6.2.15) by the expression $$(1 + 2^{j}\vert x\vert )^{n_{0}}$$. Applying the Cauchy–Schwarz inequality to $$\vert K_{j}(x)\vert \,(1 + 2^{j}\vert x\vert )^{n_{0}}$$ and $$(1 + 2^{j}\vert x\vert )^{-n_{0}+\frac{1} {4} }$$, we control the integral in (6.2.15) by the product

(6.2.17)

We now note that $$-2n_{0} + \frac{1} {2} < -n$$, and hence the second factor in (6.2.17) is equal to a constant multiple of 2−jn∕2. To estimate the first integral in (6.2.17) we use the simple fact that

We now have that the expression inside the supremum in (6.2.15) is controlled by

(6.2.18)

which, by Plancherel’s theorem, is equal to

(6.2.19)

for some constants C γ .

For multi-indices δ = (δ ₁, …, δ n ) and γ = (γ ₁, …, γ n ) we introduce the notation δ ≤ γ to mean δ j  ≤ γ j for all j = 1, …, n. For any | γ | ≤ n ₀ we use Leibniz’s rule to obtain for some constants $$C_{\delta,\gamma }$$

which inserted in (6.2.19) and combined with (6.2.18) yields (6.2.15). To obtain (6.2.16) we repeat the same argument for every derivative ∂ r K j . Since the Fourier transform of $$(\partial _{r}K_{j})(x)\,x^{\gamma }$$ is equal to a constant multiple of $$\partial ^{\gamma }\big(\xi _{r}m(\xi )\widehat{\zeta }(2^{-j}\xi )\big)$$, we observe that the extra factor 2−j in (6.2.16) can be combined with ξ r to write $$2^{-j}\partial ^{\gamma }\big(\xi _{r}m(\xi )\widehat{\zeta }(2^{-j}\xi )\big)$$ as $$\partial ^{\gamma }\big(m(\xi )\widehat{\zeta _{r}}(2^{-j}\xi )\big)$$, where $$\widehat{\zeta _{r}}(\xi ) =\xi _{r}\widehat{\zeta }(\xi )$$. The previous calculation with $$\widehat{\zeta _{r}}$$ replacing $$\widehat{\zeta }$$ can then be used to complete the proof of (6.2.16).

We now show that for all x ≠ 0, the series $$\sum _{j\in \mathbf{Z}}K_{j}(x)$$ converges to a function, which we denote by K(x). Indeed, as a consequence of (6.2.15) we have that

for any δ > 0, which implies that the function $$\sum _{j>0}\vert K_{j}(x)\vert $$ is integrable away from the origin and satisfies $$\int _{\delta \leq \vert x\vert \leq 2\delta }\sum _{j>0}\vert K_{j}(x)\vert \,dx < \infty $$. Now note that (6.2.15) also holds with $$-\frac{1} {4}$$ in place of $$\frac{1} {4}$$. Using this observation we obtain

and from this it follows that $$\int _{\delta \leq \vert x\vert \leq 2\delta }\sum _{j\leq 0}\vert K_{j}(x)\vert \,dx < \infty $$.

We conclude that the series $$\sum _{j\in \mathbf{Z}}K_{j}(x)$$ converges a.e. on $$\mathbf{R}^{n}\setminus \{0\}$$ to a function K(x) that coincides with the distribution $$W = m\,^{\vee }\,$$ on $$\mathbf{R}^{n}\setminus \{0\}$$ and satisfies

We now prove that the function $$K =\sum _{j\in \mathbf{Z}}K_{j}$$ (defined on R n ∖{0}) satisfies Hörmander’s condition. It suffices to prove that for all y ≠ 0 we have

(6.2.20)

Fix a y ∈ R n ∖{0} and pick a $$k \in \mathbf{Z}$$ such that 2−k  ≤ | y | ≤ 2−k+1. The part of the sum in (6.2.20) where j > k is bounded by

where we used (6.2.15). The part of the sum in (6.2.20) where j ≤ k is bounded by

using (6.2.16). Hörmander’s condition is satisfied for K, and we appeal to Theorem 5.​3.​3 to complete the proof of (6.2.13). □ 

Example 6.2.8.

Let m be a smooth function away from the origin that is homogeneous of imaginary order, i.e., for some fixed τ real and all λ > 0 we have

(6.2.21)

Then m is an L p Fourier multiplier for 1 < p < ∞. Indeed, differentiating both sides of (6.2.21) with respect to ∂ ξ α we obtain

and taking λ =  | ξ | ⁻¹, we deduce condition (6.2.14) with $$C_{\alpha } =\sup _{\vert \theta \vert =1}\vert \partial ^{\alpha }m(\theta )\vert $$. An explicit example of such a function is m(ξ) =  | ξ |  i τ . Another example is

which is homogeneous of degree zero and also smooth on R n ∖{0}.

Example 6.2.9.

Let z be a complex numbers with $$\mbox{ Re}\,z \geq 0$$. Then the functions

defined on R n are L p Fourier multipliers for 1 < p < ∞. To prove this assertion for m ₁, we verify condition (6.2.14). To achieve this, introduce the function on R n+1

where ξ = (ξ ₁, …, ξ n ). Then M is homogeneous of degree 0 and smooth on R n+1 ∖{0}. The derivatives ∂ β M ₁ are homogeneous of degree − | β | and by the calculation in the preceding example they satisfy $$\vert \partial ^{\beta }M_{1}(\xi,t)\vert \leq C_{\beta }\vert (\xi,t)\vert ^{-\vert \beta \vert }$$, with $$C_{\beta } =\sup _{\vert \theta \vert =1}\vert \partial ^{\beta }M_{1}(\theta )\vert $$, whenever (ξ, t) ≠ 0 and β is a multi index of n + 1 variables. In particular, taking β = (α, 0), we obtain

and setting t = 1 we deduce that $$\vert \partial ^{\alpha }m_{1}(\xi )\vert \leq C_{\alpha }(1 + \vert \xi \vert ^{2})^{-\vert \alpha \vert /2} \leq C_{\alpha }\vert \xi \vert ^{-\vert \alpha \vert }$$.

For m ₂ we introduce the function

on R n+1, which is homogeneous of degree − 2z. Then the derivative ∂ β M ₂ is homogeneous of degree − | β | − 2z, hence it satisfies $$\vert \partial ^{\beta }M_{2}(\xi,t)\vert \leq C_{\beta }\vert (\xi,t)\vert ^{-\vert \beta \vert -2\mbox{ Re}\,z}$$ for all multi-indices β of n + 1 variables. In particular, taking β = (α, 0), we obtain

and setting t = 1, we deduce $$\vert \partial ^{\alpha }m_{2}(\xi )\vert \leq C_{\alpha }(1 + \vert \xi \vert ^{2})^{-\vert \alpha \vert /2} \leq C_{\alpha }\vert \xi \vert ^{-\vert \alpha \vert }$$, where in the first inequality we used that $$\mbox{ Re}\,z \geq 0$$.

We end this section by comparing Theorems 6.2.2 and 6.2.4 with Theorem 6.2.7. It is obvious that in dimension n = 1, Theorem 6.2.2 is stronger than Theorem 6.2.7 in view of the inequality

which implies that (6.2.2) is weaker than (6.2.12). Note also that in Theorem 6.2.2 the multiplier m is not required to be differentiable at the points ± 2 j . But in higher dimensions neither theorem includes the other. In Theorem 6.2.4 the multiplier is allowed to be singular on a set of measure zero but is required to be differentiable in every variable, i.e., to be at least $$\mathcal{C}^{n}$$ in the complement of this null set. In Theorem 6.2.7, the multiplier is only allowed to be singular only at the origin, but it is assumed to be $$\mathcal{C}^{[n/2]+1}$$, requiring almost half the differentiability called for by condition (6.2.9). It should be noted that both theorems have their shortcomings. In particular, they are not L p sensitive, i.e., delicate enough to detect whether m is a bounded Fourier multiplier on some L p but not on some other L q .

Exercises

6.2.1. Let ψ(ξ) be a smooth function supported in [3∕4, 2] ∪ [−2, −3∕4] and equal to 1 on [1, 3∕2] ∪ [−3∕2, −1] that satisfies $$\sum _{j\in \mathbf{Z}}\psi (2^{-j}\xi ) = 1$$ for all ξ ≠ 0. Let 1 ≤ k ≤ n. Prove that $$m \in \mathcal{M}_{p}(\mathbf{R}^{n})$$ if and only if (6.2.1) is satisfied with m j (ξ) replaced by the function $$m(\xi )\psi (2^{-j_{1}}\xi _{1})\cdots \psi (2^{-j_{k}}\xi _{k})$$.

[Hint: To prove one direction, partition $$\mathbf{Z}^{k}$$ in 2 k sets such that for every $$\mathbf{j} = (j_{1},\ldots,j_{k})$$ in each of these sets, j i has a fixed remainder modulo 2. For the other direction, use Theorem 6.1.6 in the variables x ₁, …, x k . Also use the inequality $$\|f\|_{L^{p}(\mathbf{R}^{n})} \leq C_{p}\|(\sum _{\mathbf{j}\in \mathbf{Z}^{k}}\vert (\widehat{f}\chi _{R_{\mathbf{j}}})\,^{\vee }\,\vert ^{2})^{1/2}\|_{L^{p}(\mathbf{R}^{n})}$$, $$R_{\mathbf{j}} = ([-2,-\frac{1} {2}] \cup [\frac{1} {2},2])^{k} \times \mathbf{R}^{n-k}$$, which can be derived by duality from the identity $$\sum _{\mathbf{j}\in \mathbf{Z}^{k}}\chi _{R_{\mathbf{j}}} = 2^{k}$$.]

6.2.2. Let $$\varphi$$ be a smooth function on the real line supported in the interval [−1, 1]. Let ψ(t) be a smooth function on the real line that is equal to 1 when | t | ≥ 10 and vanishes when | t | ≤ 9. Show that for the function $$m(\xi _{1},\xi _{2}) = e^{i\xi _{2}^{2}/\xi _{ 1}}\varphi (\xi _{2})\psi (\xi _{1})$$ lies in $$\mathcal{M}_{p}(\mathbf{R}^{2})$$, $$1 < p < \infty $$, using Theorem 6.2.4. Also show that Theorem 6.2.7 does not apply.

6.2.3. Consider the differential operators

Prove that for every $$1 < p < \infty $$ there exists a constant $$C_{p} < \infty $$ such that for all Schwartz functions f on $$\mathbf{R}^{3}$$ we have

[Hint: Use Corollary 6.2.5 and the idea of Example 6.2.6.]

6.2.4. Suppose that m(ξ) is a real-valued function that satisfies either (6.2.9) or $$\vert \partial ^{\alpha }m(\xi )\vert \leq C_{\alpha }\vert \xi \vert ^{-\vert \alpha \vert }$$ for all multi-indices α with $$\vert \alpha \vert \leq [\tfrac{n} {2} ] + 1$$ and all ξ ∈ R n ∖{0}. Show that e im(ξ) lies in $$\mathcal{M}_{p}(\mathbf{R}^{n})$$ for any 1 < p < ∞.

[Hint: Prove by induction and use that

where the sum is taken over all partitions of the multi-index α as a linear combination of multi-indices β j with coefficients $$l_{j} \in \mathbf{Z}^{+} \cup \{ 0\}$$.]

6.2.5. Suppose that $$\varphi (\xi )$$ is a smooth function on R n that vanishes in a neighborhood of the origin and is equal to 1 in a neighborhood of infinity. Prove that the function $$e^{i\xi _{j}\vert \xi \vert ^{-1} }\varphi (\xi )$$ is in $$\mathcal{M}_{p}(\mathbf{R}^{n})$$ for 1 < p < ∞.

6.2.6. Let $$\tau,\tau _{1},\ldots,\tau _{n}$$ be real numbers and ρ ₁, …, ρ n be even natural numbers. Prove that the following functions are L p multipliers on R n for 1 < p < ∞:

6.2.7. Let $$\widehat{\zeta }(\xi )$$ be a smooth function on R n is supported in a compact set that does not contain the origin and let a j be a bounded sequence of complex numbers. Prove that the function

is in $$\mathcal{M}_{p}(\mathbf{R}^{n})$$ for all 1 < p < ∞.

6.2.8. Let $$\widehat{\zeta }(\xi )$$ be a smooth function on R n supported in a compact set that does not contain the origin and let $$\varDelta _{j}^{\zeta }(f) =\big (\widehat{f}(\xi )\widehat{\zeta }(2^{-j}\xi )\big)\,^{\vee }\,$$. Show that the operator

is bounded on $$L^{p}(\mathbf{R})$$ when $$1 < p < \infty $$.

[Hint: Pick a Schwartz function $$\varphi$$ satisfying $$\sum _{j\in \mathbf{Z}}\widehat{\varphi }(2^{-j}\xi ) = 1$$ on R n ∖{0} with $$\widehat{\varphi }(\xi )$$ supported in $$\frac{6} {7} \leq \vert \xi \vert \leq 2$$. Then $$\varDelta _{k}^{\varphi }\varDelta _{j}^{\zeta } = 0$$ if | j − k |  < c ₀ and we have

which is a finite sum plus a term controlled by a multiple of the operator

where M is the Hardy–Littlewood maximal function.]

6.2.9. Let Ψ be a Schwartz function whose Fourier transform is real-valued, supported in a compact set that does not contain the origin, and satisfies

Let Δ j be the Littlewood–Paley operator associated with Ψ. Prove that

as N → ∞ for all functions $$g \in \mathcal{S}(\mathbf{R}^{n})$$. Deduce that Schwartz functions whose Fourier transforms have compact supports that do not contain the origin are dense in $$L^{p}(\mathbf{R}^{n})$$ for 1 < p < ∞.

[Hint: Use the result of Exercise 6.2.8 and the Lebesgue dominated convergence theorem.]

6.3 Applications of Littlewood–Paley Theory

We now turn our attention to some important applications of Littlewood–Paley theory. We are interested in obtaining bounds for singular and maximal operators. These bounds are obtained by controlling the corresponding operators by quadratic expressions.

6.3.1 Estimates for Maximal Operators

One way to control the maximal operator $$\sup _{k}\vert T_{k}(f)\vert $$ is by introducing a good averaging function φ and using the majorization

(6.3.1)

for some constant C φ depending on φ. We apply this idea to prove the following theorem.

Theorem 6.3.1.

Let m be a bounded function on R n that is $$\mathcal{C}^{1}$$ in a neighborhood of the origin and satisfies m(0) = 1 and $$\vert m(\xi )\vert \leq C\vert \xi \vert ^{-\varepsilon }$$ for some C,ɛ > 0 and all ξ ≠ 0. For each $$k \in \mathbf{Z}$$ define $$T_{k}(f)(x) = (\widehat{f}(\xi )m(2^{-k}\xi ))\,^{\vee }\,(x)$$ . Then there is a constant C n such that for all L ² functions f on R n we have

(6.3.2)

Proof.

Select a Schwartz function φ such that $$\widehat{\varphi }(0) = 1$$. Then there are positive constants C ₁ and C ₂ such that $$\vert m(\xi ) -\widehat{\varphi } (\xi )\vert \leq C_{1}\vert \xi \vert ^{-\varepsilon }$$ for | ξ | away from zero and $$\vert m(\xi ) -\widehat{\varphi } (\xi )\vert \leq C_{2}\vert \xi \vert $$ for | ξ | near zero. These two inequalities imply that

from which the L ² boundedness of the operator

follows easily. Using estimate (6.3.1) and the well-known L ² estimate for the Hardy–Littlewood maximal function, we obtain (6.3.2). □ 

If m(ξ) is the characteristic function of a rectangle with sides parallel to the axes, this result can be extended to L p .

Theorem 6.3.2.

Let 1 < p < ∞ and let U be the n-fold product of open intervals that contain zero. For each $$k \in \mathbf{Z}$$ define $$T_{k}(f)(x) = (\widehat{f}(\xi )\chi _{U}(2^{-k}\xi ))\,^{\vee }\,(x)$$ . Then there is a constant C p,n such that for all L p functions f on R n we have

Proof.

Let us fix an open annulus A whose interior contains the boundary of U and take a smooth function with compact support $$\widehat{\psi }$$ that vanishes in a neighborhood of zero and a neighborhood of infinity and is equal to 1 on the annulus A. Then the function $$\widehat{\varphi }= (1-\widehat{\psi }\,)\chi _{U}$$ is Schwartz. Since $$\chi _{U} =\chi _{U}\widehat{\psi }+\widehat{\varphi }$$, it follows that for all $$f \in L^{p}(\mathbf{R}^{n})$$ we have

Taking the supremum over k and using Corollary 2.​1.​12 we obtain

(6.3.3)

The operator $$T_{k}(f) - f {\ast}\varphi _{2^{-k}}$$ is given by multiplication on the Fourier transform side by the multiplier

Since $$\{2^{k}U\}_{k\in \mathbf{Z}}$$ is a measurable family of rectangles with sides parallel to the axes, Exercise 5.6.1(b) yields the following inequality:

(6.3.4)

Since $$f {\ast}\psi _{2^{-k}} =\varDelta _{ j}^{\psi }(f)$$, estimate (6.1.4) of Theorem 6.1.2 yields that the expression on the right in (6.3.4) is controlled by a multiple of $$\|f\|_{L^{p}}$$. Taking L p norms in (6.3.3) and using the L p estimate for the square function yields the required conclusion. □ 

The following lacunary version of the Carleson–Hunt theorem is yet another indication of the powerful techniques of Littlewood–Paley theory.

Corollary 6.3.3.

(a) Let f be in L ² ( R n ) and let $$\varOmega$$ be an open set that contains the origin in R n . Then

for almost all x ∈ R n .

(b) Let f be in L p ( R n ) for some 1 < p < ∞. Then

for almost all x ∈ R n .

Proof.

Both limits exist everywhere for functions f in the Schwartz class. To obtain almost everywhere convergence for general f in L p we appeal to Theorem 2.​1.​14.​ The required control of the corresponding maximal operator is a consequence of Theorem 6.3.1 with $$m = \chi_{\Omega}$$ in case (a) and Theorem 6.3.2 in case (b). □ 

6.3.2 Estimates for Singular Integrals with Rough Kernels

We now turn to another application of the Littlewood–Paley theory involving singular integrals.

Theorem 6.3.4.

Suppose that μ is a finite Borel measure on R n with compact support that satisfies $$\vert \widehat{\mu }(\xi )\vert \leq B\min \big(\vert \xi \vert ^{-b},\vert \xi \vert ^{b}\big)$$ for some b > 0 and all ξ ≠ 0. Define measures μ j by setting $$\widehat{\mu _{j}}(\xi ) =\widehat{\mu } (2^{-j}\xi )$$ . Then the operator

is bounded on $$L^{p}(\mathbf{R}^{n})$$ for all 1 < p < ∞.

Proof.

It is natural to begin with the L ² boundedness of T μ . The estimate on $$\widehat{\mu }$$ implies that

(6.3.5)

The L ² boundedness of T μ is an immediate consequence of (6.3.5).

We now turn to the L p boundedness of T μ for 1 < p < ∞. We fix a radial Schwartz function ψ whose Fourier transform is supported in the annulus $$\frac{1} {2} < \vert \xi \vert < 2$$ that satisfies

(6.3.6)

whenever ξ ≠ 0. We let $$\psi _{2^{-k}}(x) = 2^{kn}\psi (2^{k}x)$$, so that $$\widehat{\psi _{2^{-k}}}(\xi ) =\widehat{\psi } (2^{-k}\xi )$$, and we observe that the identity

is valid by taking Fourier transforms and using (6.3.6). We now define operators S k by setting

Then for f in $$\mathcal{S}$$ we have that

It suffices therefore to obtain L p boundedness for the sum of the S k ’s. We begin by investigating the L ² boundedness of each S k . Since the product $$\widehat{\psi _{2^{-j-k}}}\widehat{\psi _{2^{-j'-k}}}$$ is nonzero only when j′ ∈ { j − 1, j, j + 1}, it follows that

We have therefore obtained that for all $$k \in \mathbf{Z}$$ and $$f \in \mathcal{S}(\mathbf{R}^{n})$$ we have

(6.3.7)

We notice that for any R > 0 we have

thus condition (5.​3.​4) of Theorem 5.​3.​3 is satisfied.

Next we verify that the kernel of each S k satisfies Hörmander’s condition with constant at most a multiple of (1 + | k | ). Fix y ≠ 0. Then

where

We observe that I j, k (y) ≤ C ₄ ∥ μ ∥  ℳ . Let | μ | be the total variation of μ. To obtain a more delicate estimate for I j, k (y) we argue as follows:

where | θ | ≤ 2 j+k  | y | . Note that θ depends on j, k, and y. From this and from $$I_{j,k}(y) \leq C_{4}\|\mu \|_{\mathcal{M}}$$ we obtain

(6.3.8)

which is valid for all j, k, and y ≠ 0. To estimate the last double integral even more delicately, we consider the following two cases: | x | ≥ 2 k+2 | z | and | x |  < 2 k+2 | z | . In the first case we have $$\vert x - 2^{k}z -\theta \vert \geq \frac{1} {4}\vert x\vert,$$ given the fact that | x | ≥ 2 j+k+1 | y | . In the second case we have that | x | ≤ 2 k+2 R, where B(0, R) contains the support of μ. Applying these observations in the last double integral, we obtain the following estimate:

provided 2 j  | y | ≥ 2R. Combining this estimate with (6.3.8), we obtain

(6.3.9)

We now estimate $$\sum _{j}I_{j,k}(y)$$. When 2 k  ≥ (2R)⁻¹ we use (6.3.9) to obtain

Also when 2 k  < (2R)⁻¹ we again use (6.3.9) to obtain

since in the second sum we have 2 j  | y | ≥ 2−k  > 2R, which justifies use of the corresponding estimate in (6.3.9). This gives

(6.3.10)

where the constant C ₁₃ depends on the dimension and on R. We now use estimates (6.3.7) and (6.3.10) and Theorem 5.​3.​3 to obtain that each S k maps L ¹(R n ) to L 1, ∞ (R n ) with constant at most

It follows from the Marcinkiewicz interpolation theorem (Theorem 1.​3.​2) that S k maps $$L^{p}(\mathbf{R}^{n})$$ to itself for 1 < p < 2 with bound at most $$C_{p,n}2^{-b\vert k\vert \theta _{p}}(1 + \vert k\vert )^{1-\theta _{p}}$$, when $$\frac{1} {p} = \frac{\theta _{p}} {2} + 1 -\theta _{p}$$. Summing over all $$k \in \mathbf{Z}$$, we obtain that T μ maps $$L^{p}(\mathbf{R}^{n})$$ to itself for 1 < p < 2. The boundedness of T μ for p > 2 follows by duality. □ 

An immediate consequence of the previous result is the following.

Corollary 6.3.5.

Suppose that μ is a finite Borel measure on R n with compact support that satisfies $$\vert \widehat{\mu }(\xi )\vert \leq B\min \big(\vert \xi \vert ^{-b},\vert \xi \vert ^{b}\big)$$ for some b > 0 and all ξ ≠ 0. Define measures μ j by setting $$\widehat{\mu _{j}}(\xi ) =\widehat{\mu } (2^{-j}\xi )$$ . Then the square function

(6.3.11)

maps $$L^{p}(\mathbf{R}^{n})$$ to itself whenever 1 < p < ∞.

Proof.

To obtain the boundedness of the square function in (6.3.11) we use the Rademacher functions r j (t), introduced in Appendix C.1, reindexed so that their index set is the set of all integers (not the set of nonnegative integers). For each t we introduce the operators

Next we observe that for each t in [0, 1] the operators T μ t map $$L^{p}(\mathbf{R}^{n})$$ to itself with the same constant as the operator T μ , which is in particular independent of t. Using that the square function in (6.3.11) raised to the power p is controlled by a multiple of the quantity

a fact stated in Appendix C.2, we obtain the required conclusion by integrating over R n . □ 

6.3.3 An Almost Orthogonality Principle on L p

Suppose that T j are multiplier operators given by $$T_{j}(f) = (\widehat{f}m_{j})\,^{\vee }\,$$, for some multipliers m j . If the functions m j have disjoint supports and they are bounded uniformly in j, then the operator

is bounded on L ². The following theorem gives an L p analogue of this result.

Theorem 6.3.6.

Suppose that 1 < p ≤ 2 ≤ q < ∞. Let m j be Schwartz functions supported in the annuli $$2^{j-1} \leq \vert \xi \vert \leq 2^{j+1}$$ and let $$T_{j}(f) = (\widehat{f}m_{j})\,^{\vee }\,$$ . Suppose that the T j ’s are uniformly bounded operators from L p ( R n ) to L q ( R n ), i.e.,

Then for each f ∈ L p ( R n ), the series

converges in the L q norm and there exists a constant C p,q,n < ∞ such that

(6.3.12)

Proof.

Fix a radial Schwartz function φ whose Fourier transform $$\widehat{\varphi }$$ is real, equal to one on the annulus $$\frac{1} {2} \leq \vert \xi \vert \leq 2$$, and vanishes outside the annulus $$\frac{1} {4} \leq \vert \xi \vert \leq 4$$. We set $$\varphi _{2^{-j}}(x) = 2^{jn}\varphi (2^{j}x)$$, so that $$\widehat{\varphi _{2^{-j}}}$$ is equal to 1 on the support of each m j . Setting $$\varDelta _{j}(f) = f {\ast}\varphi _{2^{-j}}$$, we observe that

for all $$j \in \mathbf{Z}$$. For a positive integer N we set

Fix $$f \in L^{p}(\mathbf{R}^{n})$$. Clearly for every N, T N (f) is in L q (R n ). Using (6.1.21) we obtain

where we used Minkowski’s inequality, since q∕2 ≥ 1. Using the uniform boundedness of the T j ’s from L p to L q , we deduce that

where we used the result of Exercise 1.1.5(b), since p ≤ 2, and Theorem 6.1.2. We conclude that the operators T N are uniformly bounded from $$L^{p}(\mathbf{R}^{n})$$ to $$L^{q}(\mathbf{R}^{n})$$.

If $$\widehat{h}$$ is compactly supported in a subset of R n ∖{0}, then the sequence T N (h) becomes independent of N for N large enough and hence it is Cauchy in L q . But in view of Exercise 6.2.9, the set of all such h is dense in L p (R n ). Combining these two results with the uniform boundedness of the T N ’s from L p to L q , a simple $$\frac{\varepsilon }{3}$$ argument gives that for all f ∈ L p the sequence T N (f) is Cauchy in L q . Therefore, for all f ∈ L p the sequence $$\{T^{N}(f)\}_{N}$$ converges in L q to some T(f). Fatou’s lemma gives

which proves (6.3.12). □ 

Exercises

6.3.1. (The g-function) Let $$P_{t}(x) = \varGamma (\frac{n+1} {2} )\pi ^{-\frac{n+1} {2} }t(t^{2} + \vert x\vert ^{2})^{-\frac{n+1} {2} }$$ be the Poisson kernel on R n .

(a) Use Exercise 6.1.4 with $$\varPsi (x) = \tfrac{\partial } {\partial t}P_{t}(x)\big\vert _{t=1}$$ to obtain that the operator

is bounded from $$L^{p}(\mathbf{R}^{n})$$ to $$L^{p}(\mathbf{R}^{n})$$ for 1 < p < ∞.

(b ) Use Exercise 6.1.4 with $$\varPsi (x) = \partial _{k}P_{1}(x)$$ to obtain that the operator

is bounded from $$L^{p}(\mathbf{R}^{n})$$ to $$L^{p}(\mathbf{R}^{n})$$ for 1 < p < ∞.

(c) Conclude that the g-function

is bounded from $$L^{p}(\mathbf{R}^{n})$$ to $$L^{p}(\mathbf{R}^{n})$$ for 1 < p < ∞.

6.3.2. Suppose that μ is a finite Borel measure on R n with compact support that satisfies $$\widehat{\mu }(0) = 0$$ and $$\vert \widehat{\mu }(\xi )\vert \leq C\vert \xi \vert ^{-a}$$ for some a > 0 and all ξ ≠ 0. Define measures μ j by setting $$\widehat{\mu _{j}}(\xi ) =\widehat{\mu } (2^{-j}\xi )$$. Show that the operator

is bounded from $$L^{p}(\mathbf{R}^{n})$$ to $$L^{p}(\mathbf{R}^{n})$$ for all 1 < p < ∞.

[Hint: Use Theorem 6.3.4]

6.3.3. ([50], [71]) (a) Suppose that μ is a finite Borel measure on R n with compact support that satisfies $$\vert \widehat{\mu }(\xi )\vert \leq C\vert \xi \vert ^{-a}$$ for some a > 0 and all ξ ≠ 0. Show that the maximal function

is bounded from L p (R n ) to $$L^{p}(\mathbf{R}^{n})$$ for all 1 < p < ∞.

(b) Let μ be the surface measure on the sphere $$\mathbf{S}^{n-1}$$ when n ≥ 2. Conclude that the dyadic spherical maximal function $$\mathcal{M}_{\mu }$$ is bounded on $$L^{p}(\mathbf{R}^{n})$$ for all 1 < p < ∞.

[Hint: Part (a): Pick a $$\mathcal{C}_{0}^{\infty }$$ function $$\varphi$$ on $$\mathbf{R}^{n}$$ with $$\widehat{\varphi }(0) = 1$$. Then the measure $$\sigma =\mu -\widehat{\mu }(0)\varphi$$ satisfies the hypotheses of Corollary 6.3.5. Since,

it follows that $$\mathcal{M}_{\mu }$$ is bounded on $$L^{p}(\mathbf{R}^{n})$$ whenever $$1 < p < \infty $$. Part (b): If $$\mu = d\sigma$$ is surface measure on $$\mathbf{S}^{n-1}$$, then $$\vert \widehat{d\sigma }(\xi )\vert \leq C\vert \xi \vert ^{-\frac{n-1} {2} }$$ (Appendices B.4 and B.7).]

6.3.4. Let $$\varOmega$$ be in $$L^{q}(\mathbf{S}^{n-1})$$ for some $$1 < q < \infty $$ and define the absolutely continuous measure

Show that for all a < 1∕q′ we have that $$\vert \widehat{\mu }(\xi )\vert \leq C\vert \xi \vert ^{-a}$$. Under the additional hypothesis that $$\varOmega$$ has mean value zero, conclude that the singular integral operator

is $$L^{p}$$ bounded for all $$1 < p < \infty $$. This provides an alternative proof of Theorem 5.​2.​10 under the hypothesis that $$\varOmega \in L^{q}(\mathbf{S}^{n-1})$$.

6.3.5. For a continuous function F on $$\mathbf{R}$$ define

Given $$f \in L_{\text{loc}}^{1}(\mathbf{R})$$ we denote by F f the indefinite integral of f, that is,

Prove that for all 1 < p < ∞ there exist constants c p and C p such that for all functions $$f \in L^{p}(\mathbf{R})$$ we have

[Hint: Let $$\varphi =\chi _{[-1,0]} -\chi _{[0,1]}$$. Then

and the double inequality follows from parts (b) and (c) of Exercise 6.1.4.]

6.3.6. Let m be a bounded function on R n that is $$\mathcal{C}^{1}$$ in a neighborhood of zero, it satisfies m(0) = 1 and $$\vert m(\xi )\vert \leq B\vert \xi \vert ^{-\varepsilon }$$ for all ξ ≠ 0, for some $$B,\varepsilon > 0$$. Define an operator T t by setting $$T_{t}(f)\,\,\widehat{\,}\,\,(\xi ) =\widehat{ f}(\xi )m(t\xi )$$. Show that the maximal operator

maps L ²(R n ) to itself.

[Hint: Majorize this maximal operator by a constant multiple of the sum

where $$\varphi$$ is a $$\mathcal{C}_{0}^{\infty }$$ function such that $$\widehat{\varphi }(0) = 1$$.]

6.3.7. ([150]) Let 0 < β < 1 and $$p_{0} = (1 -\beta /2)^{-1}$$. Suppose that $$\{f_{j}\}_{j\in \mathbf{Z}}$$ are L ² functions on the real line with norm at most 1. Assume that each f j is supported in interval of length 1 and that the orthogonality relation $$\big\vert \big\langle f_{j}\,\vert \,f_{k}\rangle \big\vert \leq (1 + \vert j - k\vert )^{-\beta }$$ holds for all $$j,k \in \mathbf{Z}$$.

(a) Let $$I \subseteq \mathbf{Z}$$ be such that for all j ∈ I the functions f j are supported in a fixed interval of length 3. Show that for all p satisfying 0 < p ≤ 2 there is $$C_{p,\beta } < \infty $$ such that

whenever ɛ j are complex numbers with | ɛ j  | ≤ 1.

(b) Under the same hypothesis as in part (a), prove that for all 0 < p < p ₀ there is a constant $$C'_{p,\beta } < \infty $$ such that

for all complex-valued sequences $$\{c_{j}\}_{j}$$ in ℓ p .

(c) Derive the conclusion of part (b) without the assumption that the f j are supported in a fixed interval of length 3.

[Hint: Part (a): Pass from L p to L ² and use the hypothesis. Part (b): Assume $$\sum _{j\in \mathbf{Z}}\vert c_{j}\vert ^{p} = 1$$. For each k = 0, 1, …, set $$I_{k} =\{ j \in \mathbf{Z}:\,\, 2^{-k-1} < \vert c_{j}\vert \leq 2^{-k}\}$$. Write $$\big\|\sum _{j\in \mathbf{Z}}c_{j}f_{j}\big\|_{L^{p}} \leq \sum _{k=0}^{\infty }2^{-k}\big\|\sum _{j\in I_{k}}(c_{j}2^{k})f_{j}\big\|_{L^{p}}$$, use part (b), Hölder’s inequality, and the fact that $$\sum _{k=0}^{\infty }2^{-kp}\vert I_{k}\vert \leq 2^{p}$$. Part (c): Write $$\sum _{j\in \mathbf{Z}}c_{j}f_{j} =\sum _{m\in \mathbf{Z}}F_{m}$$, where F m is the sum of c j f j over all j such that the support of f j meets the interval [m, m + 1]. These F m ’s are supported in [m − 1, m + 2] and are almost orthogonal.]

6.4 The Haar System, Conditional Expectation, and Martingales

There is a very strong connection between the Littlewood–Paley operators and certain notions from probability, such as conditional expectation and martingale difference operators. The conditional expectation we are concerned with is with respect to the increasing σ-algebra of all dyadic cubes on R n .

6.4.1 Conditional Expectation and Dyadic Martingale Differences

We recall the definition of dyadic cubes.

Definition 6.4.1.

A dyadic interval in $$\mathbf{R}$$ is an interval of the form

where m, k are integers. A dyadic cube in R n is a product of dyadic intervals of the same length. That is, a dyadic cube is a set of the form

for some integers m ₁, …, m n , k.

We defined dyadic intervals to be closed on the left and open on the right, so that different dyadic intervals of the same length are always disjoint sets.

Given a cube Q in R n we denote by | Q | its Lebesgue measure and by ℓ(Q) its side length. We clearly have | Q |  = ℓ(Q) n . We introduce some more notation.

Definition 6.4.2.

For $$k \in \mathbf{Z}$$ we denote by $$\mathcal{D}_{k}$$ the set of all dyadic cubes in R n whose side length is 2−k . We also denote by $$\mathcal{D}$$ the set of all dyadic cubes in $$\mathbf{R}^{n}$$. Then we have

and moreover, the $$\sigma$$-algebra $$\sigma (\mathcal{D}_{k})$$ of measurable subsets of $$\mathbf{R}^{n}$$ formed by countable unions and complements of elements of $$\mathcal{D}_{k}$$ is increasing as k increases.

We observe the fundamental property of dyadic cubes, which clearly justifies their usefulness. Any two dyadic intervals of the same side length either are disjoint or coincide. Moreover, either two given dyadic intervals are disjoint, or one contains the other. Similarly, either two dyadic cubes are disjoint, or one contains the other.

Definition 6.4.3.

Given a locally integrable function f on $$\mathbf{R}^{n}$$, we denote by

the average of f over a cube Q.

The conditional expectation of a locally integrable function f on R n with respect to the increasing family of σ-algebras $$\sigma (\mathcal{D}_{k})$$ generated by $$\mathcal{D}_{k}$$ is defined as

for all $$k \in \mathbf{Z}$$. We also define the dyadic martingale difference operator D k as follows:

also for $$k \in \mathbf{Z}$$.

Next we introduce the family of Haar functions.

Definition 6.4.4.

For a dyadic interval $$I = [m2^{-k},(m + 1)2^{-k})$$ we define $$I_{L} = [m2^{-k},(m + \frac{1} {2})2^{-k})$$ and $$I_{R} = [(m + \frac{1} {2})2^{-k},(m + 1)2^{-k})$$ to be the left and right parts of I, respectively. The function

is called the Haar function associated with the interval I.

We remark that Haar functions are constructed in such a way that they have L ² norm equal to 1. Moreover, the Haar functions have the following fundamental orthogonality property:

(6.4.1)

To see this, observe that the Haar functions have L ² norm equal to 1 by construction. Moreover, if I ≠ I′, then I and I′ must have different lengths, say we have | I′ |  <  | I | . If I and I′ are not disjoint, then I′ is contained either in the left or in the right half of I, on either of which h I is constant. Thus (6.4.1) follows.

We recall the notation

valid for square integrable functions. Under this notation, (6.4.1) can be rewritten as $$\big\langle h_{I},h_{I'}\big\rangle =\delta _{I,I'}$$, where the latter is 1 when I = I′ and zero otherwise.

6.4.2 Relation Between Dyadic Martingale Differences and Haar Functions

We have the following result relating the Haar functions to the dyadic martingale difference operators in dimension one.

Proposition 6.4.5.

For every locally integrable function f on $$\mathbf{R}$$ and for all $$k \in \mathbf{Z}$$ we have the identity

(6.4.2)

and also

(6.4.3)

Proof.

We observe that every interval J in $$\mathcal{D}_{k}$$ is either an I L or an I R for some unique $$I \in \mathcal{D}_{k-1}$$. Thus we can write

(6.4.4)

But we also have

(6.4.5)

Now taking the difference between (6.4.4) and (6.4.5) we obtain

which is easily checked to be equal to

Finally, (6.4.3) is a consequence of (6.4.1). □ 

Theorem 6.4.6.

Every function $$f \in L^{2}(\mathbf{R}^{n})$$ can be written as

(6.4.6)

where the series converges almost everywhere and in L ² . We also have

(6.4.7)

Moreover, when n = 1 we have the representation

(6.4.8)

where the sum converges a.e. and in L ² and also

(6.4.9)

Proof.

In view of the Lebesgue differentiation theorem, Corollary 2.​1.​16, given a function $$f \in L^{2}(\mathbf{R}^{n})$$ there is a set N f of measure zero on R n such that for all x ∈ R n ∖ N f we have that

whenever Q j is a sequence of decreasing cubes such that $$\bigcap _{j}\overline{Q_{j}} =\{ x\}$$. Given x in R n ∖ N f there exists a unique sequence of dyadic cubes $$Q_{j}(x) \in \mathcal{D}_{j}$$ such that $$\bigcap _{j=0}^{\infty }\overline{Q_{j}(x)} =\{ x\}$$. Then for all x ∈ R n ∖ N f we have

From this we conclude that E j (f) → f a.e. as j → ∞. We also observe that since | E j (f) | ≤ M c (f), where M c denotes the uncentered maximal function with respect to cubes, we have that $$\vert E_{j}(f) - f\vert \leq 2M_{c}(f)$$, which allows us to obtain from the Lebesgue dominated convergence theorem that E j (f) → f in L ² as j → ∞.

Next we study convergence of E j (f) as j → −∞. For a given x ∈ R n and Q j (x) as before we have that

which tends to zero as j → −∞, since the side length of each Q j (x) is 2−j . Since | E j (f) | ≤ M c (f), the Lebesgue dominated convergence theorem allows us to conclude that E j (f) → 0 in L ² as j → −∞. To obtain the conclusion asserted in (6.4.6) we simply observe that

in L ² and almost everywhere as N → ∞ and M → −∞.

To prove (6.4.7) we first observe that we can rewrite D k (f) as

(6.4.10)

Using this identity we obtain that for given integers k′ > k we have

Since k′ > k, the last integral may be nonzero only when R′ ⊊ R. If this is the case, then R′ ⊆ Q R′ for some dyadic cube $$Q_{R'} \in \mathcal{D}_{k}$$ with $$Q_{R'} \subsetneq R$$. See Figure 6.1.

Fig. 6.1

Picture of the cubes R, R′, and Q R′.

Then the function $$\chi _{Q'} - 2^{-n}\chi _{R'}$$ is supported in the cube Q R′ and the function χ Q − 2−n χ R is constant on any dyadic subcube Q of R (of half its side length) and in particular is constant on Q R′. Then

since | R′ |  = 2 n  | Q′ | . We conclude that $$\big\langle D_{k}(f),D_{k'}(f)\big\rangle = 0$$ whenever k ≠ k′, from which we easily derive (6.4.7).

Now observe that (6.4.8) is a direct consequence of (6.4.2), and (6.4.9) is a direct consequence of (6.4.3). □ 

6.4.3 The Dyadic Martingale Square Function

As a consequence of identity (6.4.7), proved in the previous subsection, we obtain that

(6.4.11)

which says that the dyadic martingale square function

is L ² bounded. It is natural to ask whether there exist L p analogues of this result, and this is the purpose of the following theorem.

Theorem 6.4.7.

For any 1 < p < ∞ there exists a constant c p,n such that for every function f in $$L^{p}(\mathbf{R}^{n})$$ we have

(6.4.12)

The lower inequality subsumes the fact that if $$\big\|S(f)\big\|_{L^{p}(\mathbf{R}^{n})} < \infty $$ , then f must be an L p function.

Proof.

Let $$\{r_{j}\}_{j}$$ be the Rademacher functions (see Appendix C.1) enumerated in such a way that their index set is the set of integers. We rewrite the upper estimate in (6.4.12) as

(6.4.13)

We prove a stronger estimate than (6.4.13), namely that for all ω ∈ [0, 1] we have

(6.4.14)

where

In view of the L ² estimate (6.4.11), we have that the operator T ω is L ² bounded with norm 1. We show that T ω is weak type (1, 1).

To show that T ω is of weak type (1, 1) we fix a function f ∈ L ¹ and α > 0. We apply the Calderón–Zygmund decomposition (Theorem 5.​3.​1) to f at height α to write

where Q j are dyadic cubes that satisfy $$\sum _{j}\vert Q_{j}\vert \leq \frac{1} {\alpha } \big\|f\big\|_{L^{1}}$$ and g has L ² norm at most $$(2^{n}\alpha \big\|f\big\|_{L^{1}})^{\frac{1} {2} }$$; see (5.​3.​1). To achieve this decomposition, we apply the proof of Theorem 5.​3.​1 starting with a dyadic mesh of large cubes such that $$\vert Q\vert \geq \frac{1} {\alpha } \big\|f\big\|_{L^{1}}$$ for all Q in the mesh. Then we subdivide each Q in the mesh by halving each side, and we select those cubes for which the average of f over them is bigger than α (and thus at most 2 n α). Since the original mesh consists of dyadic cubes, the stopping-time argument of Theorem 5.​3.​1 ensures that each selected cube is dyadic.

We observe (and this is the key observation) that T ω (b) is supported in $$\bigcup _{j}Q_{j}$$. To see this, we use identity (6.4.10) to write T ω (b) as

(6.4.15)

We consider the following three cases for the cubes Q that appear in the inner sum in (6.4.15): (i) Q j  ⊆ Q, (ii) Q j ∩ Q = ∅, and (iii) $$Q \subsetneq Q_{j}$$. It is simple to see that in cases (i) and (ii) we have $$\mathop{\mbox{ Avg}}_{Q}[(f -\mathop{\mbox{ Avg}}_{Q_{j}}f)\chi _{Q_{j}}] = 0$$. Therefore the inner sum in (6.4.15) is taken over all Q that satisfy $$Q \subsetneq Q_{j}$$. But then we must have that the unique dyadic parent R of Q is also contained in Q j . It follows that the expression inside the square brackets in (6.4.15) is supported in R and therefore in Q j . We conclude that T ω (b) is supported in $$\bigcup _{j}Q_{j}$$. Using Exercise 5.3.5(a) we obtain that T ω is weak type (1, 1) with norm at most

We have now established that T ω is weak type (1, 1). Since T ω is L ² bounded with norm 1, it follows by interpolation that T ω is L p bounded for all 1 < p < 2. The L p boundedness of T ω for the remaining p > 2 follows by duality. (Note that the operators D k and E k are self-transpose.) We conclude the validity of (6.4.14), which implies that of (6.4.13). As observed, this is equivalent to the upper estimate in (6.4.12).

Finally, we notice that the lower estimate in (6.4.12) is a consequence of the upper estimate as in the case of the Littlewood–Paley operators Δ j . Indeed, we need to observe that in view of (6.4.6) we have

Taking the supremum over all functions g on R n with L p′ norm at most 1, we obtain that f gives rise to a bounded linear functional on L p′. It follows by the Riesz representation theorem that f must be an L p function that satisfies the lower estimate in (6.4.12). □ 

6.4.4 Almost Orthogonality Between the Littlewood–Paley Operators and the Dyadic Martingale Difference Operators

Next, we discuss connections between the Littlewood–Paley operators Δ j and the dyadic martingale difference operators D k . It turns out that these operators are almost orthogonal in the sense that the L ² operator norm of the composition D k Δ j decays exponentially as the indices j and k get farther away from each other.

For the purposes of the next theorem we define the Littlewood–Paley operators Δ j as convolution operators with the function $$\varPsi _{2^{-j}}$$, where

and Φ is a fixed radial Schwartz function whose Fourier transform $$\widehat{\varPhi }$$ is real-valued, supported in the ball | ξ |  < 2, and equal to 1 on the ball | ξ |  < 1. In this case we clearly have the identity

Then we have the following theorem.

Theorem 6.4.8.

There exists a constant C such that for every k,j in Z the following estimate on the operator norm of $$D_{k}\varDelta _{j}:\, L^{2}(\mathbf{R}^{n}) \rightarrow L^{2}(\mathbf{R}^{n})$$ is valid:

(6.4.16)

Proof.

Since Ψ is a radial function, it follows that Δ j is equal to its transpose operator on L ². Moreover, the operator D k is also equal to its transpose. Thus

and it therefore suffices to prove only that

(6.4.17)

By a simple dilation argument it suffices to prove (6.4.17) when k = 0. In this case we have the estimate

and since the D k ’s and Δ j ’s are self-transposes, we have

Estimate (6.4.17) when k = 0 will be a consequence of the pair of inequalities

(6.4.18)

(6.4.19)

We start by proving (6.4.18). We consider only the term E ₀ Δ j −Δ j , since the term E ₋₁ Δ j −Δ j is similar. Let f ∈ L ²(R n ). Then

where ξ x, t lies on the line segment joining x and t. Applying the Cauchy-Schwarz inequality to the first two terms, we see that the last expression is bounded by

which is clearly controlled by $$C(2^{jn} + 2^{2j})\|f\|_{L^{2}}^{2} \leq 2\,C2^{j}\|f\|_{L^{2}}^{2}$$. This proves (6.4.18).

We now turn to the proof of (6.4.19). We set $$S_{j} =\sum _{k\leq j}\varDelta _{k}$$. Since Δ j is the difference of two S j ’s, it suffices to prove (6.4.19), where Δ j is replaced by S j . We work only with the term S j E ₀ − E ₀, since the other term can be treated similarly. We have

Since the functions appearing inside the sum in the first term have supports with bounded overlap, we obtain

and the crucial observation is that

a consequence of Plancherel’s identity and the fact that $$\vert 1 -\widehat{\varPhi } (2^{-j}\xi )\vert \leq \chi _{\vert \xi \vert \geq 2^{j}}$$. Putting these observations together, we deduce

and the required conclusion will be proved if we can show that

(6.4.20)

We prove (6.4.20) by using an estimate based purely on size. Let c Q be the center of the dyadic cube Q. For x∉3Q we have the estimate

since both 2 j  ≥ 1, and | x − c Q  | ≥ 1. We now control the left-hand side of (6.4.20) by

By taking M large enough, we obtain (6.4.20) and thus (6.4.19). □ 

Exercises

6.4.1. (a) Prove that no dyadic cube in R n contains the point 0 in its interior. (b) Prove that every interval [a, b] is contained in the union of three dyadic intervals of length less than b − a. (c) Prove that every cube of length l in R n is contained in the union of 3 n dyadic cubes, each having length less than l.

6.4.2. Let $$k \in \mathbf{Z}$$. Show that the set [m2−k , (m + s)2−k ) is a dyadic interval if and only if s = 2 p for some $$p \in \mathbf{Z}$$ and m is an integer multiple of s.

6.4.3. Given a cube Q in R n of side length ℓ(Q) ≤ 2 k−1 for some integer k, prove that there is a dyadic cube D Q of side length 2 k such that $$Q \subseteq \sigma +D_{Q}$$ for some $$\sigma = (\sigma _{1},\ldots,\sigma _{n})$$, where $$\sigma _{j} \in \{ 0,1/3,-1/3\}$$.

6.4.4. Show that the martingale maximal function $$f\mapsto \sup _{k\in \mathbf{Z}}\vert E_{k}(f)\vert $$ is weak type (1, 1) with constant at most 1.

[Hint: Use Exercise 2.1.12.]

6.4.5. (a) Show that E N (f) → f a.e. as N → ∞ for all $$f \in L_{\text{loc}}^{1}(\mathbf{R}^{n})$$.

(b) Prove that E N (f) → f in L p as N → ∞ for all f ∈ L p (R n ) whenever 1 < p < ∞.

6.4.6. (a) Let $$k,k' \in \mathbf{Z}$$ be such that k ≠ k′. Show that for functions f and g in L ²(R n ) we have

(b) Conclude that for functions f j in L ²(R n ) we have

(c) Let Δ j and C be as in the statement of Theorem 6.4.8. Show that for any r ∈ Z we have

6.4.7. ([133]) Let D j , Δ j be as in Theorem 6.4.8.

(a) Prove that the operator

is bounded from L ²(R n ) to itself with norm at most a multiple of $$2^{-\frac{1} {2} \vert r\vert }$$.

(b) Show that V r is $$L^{p}(\mathbf{R}^{n})$$ bounded for all $$1 < p < \infty $$ with a constant depending only on p and n.

(c) Conclude that for each 1 < p < ∞ there is a constant c p  > 0 such that V r is bounded on $$L^{p}(\mathbf{R}^{n})$$ with norm at most a multiple of $$2^{-c_{p}\,\vert r\vert }$$.

[Hint: Part (a): Write $$\varDelta _{j} =\varDelta _{j}\widetilde{\varDelta }_{j}$$, where $$\widetilde{\varDelta }_{j}$$ is another family of Littlewood–Paley operators and use Exercise 6.4.6 (b). Part (b): Use duality and (6.1.21).]

6.5 The Spherical Maximal Function

In this section we discuss yet another consequence of the Littlewood–Paley theory, the boundedness of the spherical maximal operator.

6.5.1 Introduction of the Spherical Maximal Function

We denote throughout this section by d σ the normalized Lebesgue measure on the sphere S n−1. For f in $$L^{p}(\mathbf{R}^{n})$$, $$1 \leq p \leq \infty $$, we define the maximal operator

(6.5.1)

and we observe that by Minkowski’s integral inequality each expression inside the supremum in (6.5.1) is well defined for $$f \in L^{p}$$ for almost all $$x \in \mathbf{R}^{n}$$. The operator $$\mathcal{M}$$ is called the spherical maximal function . It is unclear at this point for which functions f we have $$\mathcal{M}(f) < \infty $$ a.e. and for which values of $$p < \infty $$ the maximal inequality

(6.5.2)

holds for all functions $$f \in L^{p}(\mathbf{R}^{n})$$.

Spherical averages often make their appearance as solutions of partial differential equations. For instance, the spherical average

(6.5.3)

is a solution of the wave equation

in $$\mathbf{R}^{3}$$. The introduction of the spherical maximal function is motivated by the fact that the related spherical average

(6.5.4)

solves Darboux’s equation

in $$\mathbf{R}^{3}$$. It is rather remarkable that the Fourier transform can be used to study almost everywhere convergence for several kinds of maximal averaging operators such as the spherical averages in (6.5.4). This is achieved via the boundedness of the corresponding maximal operator; the maximal operator controlling the averages over $$\mathbf{S}^{n-1}$$ is given in (6.5.1).

Before we begin the analysis of the spherical maximal function, we recall that

as shown in Appendix B.4. Using the estimates in Appendices B.6 and B.7 and the identity

derived in Appendix B.2, we deduce the crucial estimate

(6.5.5)

Theorem 6.5.1.

Let n ≥ 3. For each $$\frac{n} {n-1} < p \leq \infty $$ , there is a constant C p such that

(6.5.6)

holds for all f in $$L^{p}(\mathbf{R}^{n})$$ . Consequently, for all $$\frac{n} {n-1} < p \leq \infty $$ and $$f \in L^{p}(\mathbf{R}^{n})$$ we have

(6.5.7)

for almost all $$x \in \mathbf{R}^{n}$$ . Here we set $$\omega _{n-1} = \vert \mathbf{S}^{n-1}\vert $$ .

The proof of this theorem is given in the rest of this section. Before we present the proof we explain the validity of (6.5.7). Clearly this assertion is valid for functions $$f \in \mathcal{S}(\mathbf{R}^{n})$$. Using inequality (6.5.6) and Theorem 2.​1.​14 we obtain that (6.5.7) holds for all functions in $$f \in L^{p}(\mathbf{R}^{n})$$.

We now focus on (6.5.6). Define $$m(\xi ) =\widehat{ d\sigma }(\xi )$$ and notice that m(ξ) is a $$\mathcal{C}^{\infty }$$ function. To study the maximal multiplier operator

we decompose the multiplier m(ξ) into radial pieces as follows: We fix a radial $$\mathcal{C}^{\infty }$$ function φ ₀ in R n such that $$\varphi _{0}(\xi ) = 1$$ when | ξ | ≤ 1 and $$\varphi _{0}(\xi ) = 0$$ when | ξ | ≥ 2. For j ≥ 1 we let

(6.5.8)

and we observe that $$\varphi _{j}(\xi )$$ is localized near $$\vert \xi \vert \approx 2^{j}$$. Then we have

Set $$m_{j} =\varphi _{j}m$$ for all j ≥ 0. The m j ’s are $$\mathcal{C}_{0}^{\infty }$$ functions that satisfy

Also, the following estimate is valid:

where

Since the function m ₀ is $$\mathcal{C}_{0}^{\infty }$$, we have that $$\mathcal{M}_{0}$$ maps $$L^{p}$$ to itself for all 1 < p ≤ ∞. (See Exercise 6.5.1.)

We define g-functions associated with m j as follows:

where $$A_{j,t}(f)(x) =\big (\widehat{f}(\xi )\,m_{j}(t\xi )\big)\,^{\vee }\,(x)$$.

6.5.2 The First Key Lemma

We have the following lemma:

Lemma 6.5.2.

There is a constant C = C(n) < ∞ such that for any j ≥ 1 we have the estimate

for all functions f in L ² ( R n ).

Proof.

We define a function

we let $$\widetilde{A}_{j,t}(f)(x) =\big (\widehat{f}(\xi )\,\widetilde{m}_{j}(t\xi )\big)\,^{\vee }\,(x)$$, and we let

be the associated g-function. For $$f \in L^{2}(\mathbf{R}^{n})$$, the identity

is clearly valid for all j and s. Since $$A_{j,s}(f) = f {\ast} (m_{j}\,^{\vee }\,)_{s}$$ and $$m_{j}\,^{\vee }\,$$ has integral zero for j ≥ 1 (here $$(m_{j}\,^{\vee }\,)_{s}(x) = s^{-n}m_{j}\,^{\vee }\,(s^{-1}x)$$), it follows from Corollary 2.​1.​19 that

for all x ∈ R n ∖ E f , where E f is some set of Lebesgue measure zero. By the fundamental theorem of calculus for x ∈ R n ∖ E f we deduce that

from which we obtain the estimate

(6.5.9)

Taking the supremum over all t > 0 on the left-hand side in (6.5.9) and integrating over R n , we obtain the estimate

by applying the Cauchy–Schwarz inequality twice. Next we claim that as a consequence of (6.5.5) we have for some $$c,\widetilde{c} < \infty $$,

Using these facts together with the facts that the functions m j and $$\widetilde{m}_{j}$$ are supported in the annuli 2 j−1 ≤ | ξ | ≤ 2 j+1, we obtain that the g-functions G j and $$\widetilde{G}_{j}$$ are L ² bounded with norms at most a constant multiple of the quantities $$2^{-j\frac{n-1} {2} }$$ and $$2^{j(1-\frac{n-1} {2} )}$$, respectively; see Exercise 6.5.2. Note that since n ≥ 3, both exponents are negative. We conclude that

which is what we needed to prove. □ 

6.5.3 The Second Key Lemma

Next we need the following lemma.

Lemma 6.5.3.

There exists a constant C = C(n) < ∞ such that for all j ≥ 1 and for all f in L ¹ ( R n ) we have

Proof.

Let $$K^{(j)} = (\varphi _{j})\,^{\vee }\,{\ast} d\sigma =\varPhi _{2^{-j}} {\ast} d\sigma$$, where Φ is a Schwartz function. Setting

we have that

(6.5.10)

The proof of the lemma is based on the estimate:

(6.5.11)

and the weak type (1, 1) boundedness of the Hardy–Littlewood maximal operator $$\mathcal{M}$$ (Theorem 2.​1.​6). To establish (6.5.11), it suffices to show that for any M > n there is a constant C M  < ∞ such that

(6.5.12)

Then Theorem 2.​1.​10 yields (6.5.11) and hence the required conclusion.

Using the fact that Φ is a Schwartz function, we have for every N > 0,

We pick an N to depend on M (6.5.12); in fact, any N > M suffices for our purposes. We split the last integral into the regions

and for r ≥ 0,

The key observation is that whenever B(y, R) is a ball of radius R in R n centered at $$y \in \mathbf{S}^{n-1}$$, then the spherical measure of the set $$\mathbf{S}^{n-1} \cap B(y,R)$$ is at most a dimensional constant multiple of R n−1. This implies that the spherical measure of each S r (x) is at most c n 2(r+1−j)(n−1), an estimate that is useful only when r ≤ j. Using this observation, together with the fact that for y ∈ S r (x) we have | x | ≤ 2 r+1−j + 1, we obtain the following estimate for the expression $$\vert (\varPhi _{2^{-j}} {\ast} d\sigma )(x)\vert $$:

where we used that N > M > n. This establishes (6.5.12). □ 

6.5.4 Completion of the Proof

It remains to combine the previous ingredients to complete the proof of the theorem. Interpolating between the L ² → L ² and $$L^{1} \rightarrow L^{1,\infty }$$ estimates obtained in Lemmas 6.5.2 and 6.5.3, we obtain

for all 1 < p ≤ 2. When $$p > \frac{n} {n-1}$$ the series $$\sum _{j=1}^{\infty }2^{(\frac{n} {p} -(n-1))j}$$ converges and we conclude that $$\mathcal{M}$$ is $$L^{p}$$ bounded for these p’s. The boundedness of $$\mathcal{M}$$ on $$L^{p}$$ for p > 2 follows by interpolation between L q for q < 2 and the estimate $$\mathcal{M}:\, L^{\infty }\rightarrow L^{\infty }$$.

Exercises

6.5.1. Let m be in $$L^{1}(\mathbf{R}^{n}) \cap L^{\infty }(\mathbf{R}^{n})$$ that satisfies $$\vert m\,^{\vee }\,(x)\vert \leq C(1 + \vert x\vert )^{-n-\delta }$$ for some δ > 0. Show that the maximal multiplier

is L p bounded for all 1 < p < ∞.

6.5.2. Suppose that the function m is supported in the annulus R ≤ | ξ | ≤ 2R and is bounded by A. Show that the g-function

maps L ²(R n ) to L ²(R n ) with bound at most $$A\sqrt{\log 2}$$.

6.5.3. ([302]) Let A, a, b > 0 with a + b > 1. Use the idea of Lemma 6.5.2 to show that if m(ξ) satisfies | m(ξ) | ≤ A (1 + | ξ | )−a and | ∇m(ξ) | ≤ A (1 + | ξ | )−b for all ξ ∈ R n , then the maximal operator

is bounded from L ²(R n ) to itself.

[Hint: Use that

where $$\mathcal{M}_{m,j}$$ corresponds to the multiplier φ j m; here φ j is as in (6.5.8). Show that

where $$\widetilde{m}(\xi ) =\xi \cdot \nabla m(\xi )$$.]

6.5.4. Let A, c > 0, a > 1∕2, 0 < b < n. Follow the idea of the proof of Theorem 6.5.1 to obtain the following more general result: If d μ is a finite Borel measure supported in the closed unit ball that satisfies $$\vert \widehat{d\mu }(\xi )\vert \leq A\,(1 + \vert \xi \vert )^{-a}$$ for all ξ ∈ R n and d μ(B(y, R)) ≤ c R b for all R > 0, then the maximal operator

maps $$L^{p}(\mathbf{R}^{n})$$ to itself when $$p > \frac{2n-2b+2a-1} {n-b+2a-1}$$.

[Hint: Using the notation of the preceding exercise, show that $$\|\mathcal{M}_{m,j}(f)\|_{L^{2}} \leq C\,2^{j(\frac{1} {2} -a)}\|f\|_{L^{2}}$$ and that $$\|\mathcal{M}_{m,j}(f)\|_{L^{1,\infty }} \leq C\,2^{j(n-b)}\|f\|_{L^{1}}$$ for all $$j \in \mathbf{Z}^{+}$$, where C is a constant depending on the given parameters.]

6.5.5. Show that Theorem 6.5.1 is false when n = 1, that is, show that the maximal operator

is unbounded on $$L^{p}(\mathbf{R})$$ for all $$p < \infty $$.

6.5.6. Show that when n ≥ 2 and $$p \leq \frac{n} {n-1}$$ there exists an $$L^{p}(\mathbf{R}^{n})$$ function f such that $$\mathcal{M}(f)(x) = \infty $$ for all x ∈ R n . Hence Theorem 6.5.1 is false is this case.

[Hint: Choose a compactly supported and radial function equal to $$\vert y\vert ^{1-n}(-\log \vert y\vert )^{-1}$$ when | y | ≤ 1∕2.]

6.6 Wavelets and Sampling

In this section we construct orthonormal bases of $$L^{2}(\mathbf{R})$$ generated by translations and dilations of a single function. An example of such base is given by the Haar functions we encountered in Section 6.4. The Haar functions are generated by integer translations and dyadic dilations of the single function $$\chi _{ [0,\frac{1} {2} )} -\chi _{[\frac{1} {2},1)}$$. This function is not smooth, and the main question addressed in this section is whether there exist smooth analogues of the Haar functions.

Definition 6.6.1.

A square integrable function φ on R n is called a wavelet if the family of functions

where ν ranges over $$\mathbf{Z}$$ and k over Z n , is an orthonormal basis of L ²(R n ). This means that the functions $$\varphi _{\nu,k}$$ are mutually orthogonal and span L ²(R n ), and φ is normalized to have L ² norm equal to 1. Note that the Fourier transform of $$\varphi _{\nu,k}$$ is given by

(6.6.1)

Rephrasing the question posed earlier, the main issue addressed in this section is whether smooth wavelets actually exist. Before we embark on this topic, we recall that we have already encountered examples of nonsmooth wavelets.

Example 6.6.2.

(The Haar wavelet) Recall the family of functions

where I ranges over $$\mathcal{D}$$ (the set of all dyadic intervals) and I L is the left part of I and I R is the right part of I. Note that if $$I = [2^{-\nu }k,2^{-\nu }(k + 1))$$, then

where

(6.6.2)

The single function $$\varphi$$ in (6.6.2) therefore generates the Haar basis by taking translations and dilations. Moreover, we observed in Section 6.4 that the family $$\{h_{I}\}_{I}$$ is orthonormal. Moreover, in Theorem 6.4.6 we obtained the representation

which proves the completeness of the system $$\{h_{I}\}_{I\in \mathcal{D}}$$ in $$L^{2}(\mathbf{R})$$.

6.6.1 Some Preliminary Facts

Before we look at more examples, we make some observations. We begin with the following useful fact.

Proposition 6.6.3.

Let $$g \in L^{1}(\mathbf{R}^{n})$$ . Then

if and only if

for almost all x ∈ T n .

Proof.

We define the periodic function

which is easily shown to be in L ¹(T n ). Moreover, we have

for all m ∈ Z n , where $$\widehat{G}(m)$$ denotes the mth Fourier coefficient of G and $$\widehat{g}(m)$$ denotes the Fourier transform of g at ξ = m. If $$\widehat{g}(m) = 0$$ for all m ∈ Z n ∖{0}, then all the Fourier coefficients of G (except for m = 0) vanish, which means that the sequence $$\{\widehat{G}(m)\}_{m\in \mathbf{Z}^{n}}$$ lies in ℓ ¹(Z n ) and hence Fourier inversion applies. We conclude that for almost all x ∈ T n we have

Conversely, if G is a constant, then $$\widehat{G}(m) = 0$$ for all m ∈ Z n ∖{0}, and so the same holds for g. □ 

A consequence of the preceding proposition is the following.

Proposition 6.6.4.

Let $$\varphi \in L^{2}(\mathbf{R}^{n})$$ . Then the sequence

(6.6.3)

forms an orthonormal set in L ² ( R n ) if and only if

(6.6.4)

for almost all ξ ∈ R n .

Proof.

Observe that either (6.6.4) or the hypothesis that the sequence in (6.6.3) is orthonormal implies that $$\|\varphi \|_{L^{2}} = 1$$. Also the orthonormality condition

is equivalent to

in view of Parseval’s identity. Proposition 6.6.3 with $$g(\xi ) = \vert \widehat{\varphi }(\xi )\vert ^{2}$$ gives that the latter is equivalent to

for almost all ξ ∈ R n . □ 

Corollary 6.6.5.

Let $$\varphi \in L^{2}(\mathbf{R}^{n})$$ and suppose that the sequence

(6.6.5)

forms an orthonormal set in L ² ( R n ). Then the measure of the support of $$\widehat{\varphi }$$ is at least 1, that is,

(6.6.6)

Moreover, if $$\vert \mbox{ supp }\widehat{\varphi }\vert = 1$$ , then $$\vert \widehat{\varphi }(\xi )\vert = 1$$ for almost all $$\xi \in \mbox{ supp }\!\widehat{\varphi }$$ .

Proof.

It follows from (6.6.4) that $$\vert \widehat{\varphi }\vert \leq 1$$ for almost all ξ ∈ R n and thus

If equality holds in (6.6.6), then equality holds in the preceding inequality, and since $$\vert \widehat{\varphi }\vert \leq 1$$ a.e., it follows that $$\vert \widehat{\varphi }(\xi )\vert = 1$$ for almost all ξ in $$\mbox{ supp }\!\widehat{\varphi }$$. □ 

6.6.2 Construction of a Nonsmooth Wavelet

Having established these preliminary facts, we now start searching for examples of wavelets. It follows from Corollary 6.6.5 that the support of the Fourier transform of a wavelet must have measure at least 1. It is reasonable to ask whether this support can have measure exactly 1. Example 6.6.6 indicates that this can indeed happen. As dictated by the same corollary, the Fourier transform of such a wavelet must satisfy $$\vert \widehat{\varphi }(\xi )\vert = 1$$ for almost all $$\xi \in \mbox{ supp }\!\widehat{\varphi }$$, so it is natural to look for a wavelet $$\varphi$$ such that $$\widehat{\varphi }=\chi _{A}$$ for some set A. We can start by asking whether the function

on R is an appropriate Fourier transform of a wavelet, but a moment’s thought shows that the functions $$\varphi _{\mu,0}$$ and $$\varphi _{\nu,0}$$ cannot be orthogonal to each other when μ ≠ 0. The problem here is that the Fourier transforms of the functions $$\varphi _{\nu,k}$$ cluster near the origin and do not allow for the needed orthogonality. We can fix this problem by considering a function whose Fourier transform vanishes near the origin. Among such functions, a natural candidate is

(6.6.7)

which is indeed the Fourier transform of a wavelet.

Example 6.6.6.

Let $$A = [-1,-\frac{1} {2})\bigcup [\frac{1} {2},1)$$ and define a function $$\varphi$$ on $$\mathbf{R}$$ by setting

Then we assert that the family of functions

is an orthonormal basis of $$L^{2}(\mathbf{R})$$ (i.e., the function φ is a wavelet). This is an example of a wavelet with minimally supported frequency.

To verify this assertion, first note that $$\{\varphi _{0,k}\}_{k\in \mathbf{Z}}$$ is an orthonormal set, since (6.6.4) is easily seen to hold. Dilating by 2 ν , it follows that $$\{\varphi _{\nu,k}\}_{k\in \mathbf{Z}}$$ is also an orthonormal set for every fixed $$\nu \in \mathbf{Z}$$. Next, observe that if μ ≠ ν, then

(6.6.8)

This implies that the family $$\{2^{\nu /2}\varphi (2^{\nu }x - k)\}_{k\in \mathbf{Z},\nu \in \mathbf{Z}}$$ is also orthonormal.

Next, we observe that the completeness of $$\{\varphi _{\nu,k}\}_{\nu,k\in \mathbf{Z}}$$ is equivalent to that of $$\{\widehat{\varphi _{\nu,k}}(\xi )\}_{\nu,k\in \mathbf{Z}} =\{ 2^{-\nu /2}e^{-2\pi ik\xi 2^{-\nu } }\chi _{2^{\nu }A}(\xi )\}_{\nu,k\in \mathbf{Z}}$$. Let $$f \in L^{2}(\mathbf{R})$$, fix any $$\nu \in \mathbf{Z}$$, and define

Suppose that for all $$k \in \mathbf{Z}$$,

Exercise 6.6.1(a) shows $$\{e^{-2\pi ik\xi }\}_{k\in \mathbf{Z}}$$ is an orthonormal basis of L ²(A), and therefore $$\chi _{A}h = 0$$ almost everywhere. From the definition of h it follows that $$\chi _{2^{\nu }A}f = 0$$ almost everywhere. Now suppose for all $$\nu,k \in \mathbf{Z}$$

Then $$\chi _{2^{\nu }A}f = 0$$ almost everywhere for all $$\nu \in \mathbf{Z}$$. Since $$\cup _{\nu \in \mathbf{Z}}2^{\nu }A = \mathbf{R}\setminus \{0\}$$, it follows that f = 0 almost everywhere. We conclude $$\{\widehat{\varphi _{\nu,k}}\}_{\nu,k\in \mathbf{Z}}$$ is complete.

6.6.3 Construction of a Smooth Wavelet

The wavelet basis of L ²(R n ) constructed in Example 6.6.6 is forced to have slow decay at infinity, since the Fourier transforms of the elements of the basis are nonsmooth. Smoothing out the function $$\widehat{\varphi }$$ but still expecting φ to be wavelet is a bit tricky, since property (6.6.8) may be violated when μ ≠ ν, and moreover, (6.6.4) may be destroyed. These two obstacles are overcome by the careful construction of the next theorem.

Theorem 6.6.7.

There exists a Schwartz function $$\varphi$$ on the real line that is a wavelet, that is, the collection of functions $$\{\varphi _{\nu,k}\}_{k,\nu \in \mathbf{Z}}$$ with $$\varphi _{\nu,k}(x) = 2^{ \frac{\nu }{ 2} }\varphi (2^{\nu }x - k)$$ is an orthonormal basis of $$L^{2}(\mathbf{R})$$ . Moreover, the function $$\varphi$$ can be constructed so that its Fourier transform satisfies

(6.6.9)

Note that in view of condition (6.6.9), the function $$\varphi$$ must have vanishing moments of all orders.

Proof.

We start with an odd smooth real-valued function Θ on the real line such that $$\varTheta (t) = \frac{\pi } {4}$$ for $$t \geq \frac{1} {6}$$ and such that Θ is strictly increasing on the interval $$\big[-\frac{1} {6}, \frac{1} {6}\big]$$. We set

and we observe that

and that

for all real t. Next we introduce the smooth function ω defined via

on the interval $$\big[-\frac{4} {3},-\frac{1} {3}\big]\bigcup \big[\frac{1} {3}, \frac{4} {3}\big]$$. Note that ω is an even function. Finally we define the function φ by letting

and we note that

It follows that the function φ is symmetric about the number $$\frac{1} {2}$$, that is, we have

for all $$x \in \mathbf{R}$$. Note that $$\varphi$$ is a Schwartz function whose Fourier transform is supported in the set $$\big[-\tfrac{4} {3},-\tfrac{1} {3}\big]\bigcup \big[\tfrac{1} {3}, \tfrac{4} {3}\big]$$.

Having defined $$\varphi$$, we proceed by showing that it is a wavelet. In view of identity (6.6.1) we have that $$\widehat{\varphi _{\nu,k}}$$ is supported in the set $$\tfrac{1} {3}2^{\nu } \leq \vert \xi \vert \leq \tfrac{4} {3}2^{\nu }$$, while $$\widehat{\varphi _{\mu,j}}$$ is supported in the set $$\tfrac{1} {3}2^{\mu } \leq \vert \xi \vert \leq \tfrac{4} {3}2^{\mu }$$. The intersection of these sets has measure zero when | μ −ν | ≥ 2, which implies that such wavelets are orthogonal to each other. Therefore, it suffices to verify orthogonality between adjacent scales (i.e., when ν = μ and ν = μ + 1).

We begin with the case ν = μ, which, by a simple dilation, is reduced to the case ν = μ = 0. Thus to obtain the orthogonality of the functions $$\varphi _{0,k}(x) =\varphi (x - k)$$ and $$\varphi _{0,j}(x) =\varphi (x - j)$$, in view of Proposition 6.6.4, it suffices to show that

(6.6.10)

Since the sum in (6.6.10) is 1-periodic, we check that is equal to 1 only for ξ in $$\big[\frac{1} {3}, \frac{4} {3}\big]$$. First for $$\xi \in \big [\frac{1} {3}, \frac{2} {3}\big]$$, the sum in (6.6.10) is equal to

from the definition of ω. A similar argument also holds for $$\xi \in \big [\frac{2} {3}, \frac{4} {3}\big]$$, and this completes the proof of (6.6.10). As a consequence of this identity we also obtain that the functions $$\varphi _{0,k}$$ have L ² norm equal to 1, and thus so have the functions $$\varphi _{\nu,k}$$, via a change of variables.

Next we prove the orthogonality of the functions $$\varphi _{\nu,k}$$ and $$\varphi _{\nu +1,j}$$ for general $$\nu,k,j \in \mathbf{Z}$$. We begin by observing the validity of the following identity:

(6.6.11)

Indeed, from the definition of $$\varphi$$, it follows that

This function is supported in

and on this set it is equal to

by the definition of ω. This establishes (6.6.11).

We now turn to the orthogonality of the functions $$\varphi _{\nu,k}$$ and $$\varphi _{\nu +1,j}$$ for general $$\nu,k,j \in \mathbf{Z}$$. Using (6.6.1) and (6.6.11) we have

where the last identity follows from the change of variables ξ = ξ′ − 2 in the second-to-last integral, which transforms its range of integration to $$\big[\tfrac{2} {3}, \tfrac{4} {3}\big]$$ and its integrand to the negative of that of the last displayed integral.

Our final task is to show that the orthonormal system $$\{\varphi _{\nu,k}\}_{\nu,k\in \mathbf{Z}}$$ is complete. We show this by proving that whenever a square-integrable function f satisfies

(6.6.12)

for all $$\nu,k \in \mathbf{Z}$$, then f must be zero. Suppose that (6.6.12) holds. Plancherel’s identity yields

for all ν, k and thus

(6.6.13)

for all $$\nu,k \in \mathbf{Z}$$. It follows from Proposition 6.6.3 and (6.6.13) (with k = 0) that

for all $$\nu \in \mathbf{Z}$$.

Next, we show that the identity

(6.6.14)

for all $$\nu \in \mathbf{Z}$$ implies that $$\widehat{f}$$ is identically equal to zero. Suppose that $$\frac{1} {3} \leq \xi \leq \frac{2} {3}$$. In this case the support properties of $$\widehat{\varphi }$$ imply that the only terms in the sum in (6.6.14) that do not vanish are k = 0 and k = −1. Thus for $$\frac{1} {3} \leq \xi \leq \frac{2} {3}$$ the identity in (6.6.14) reduces to

hence

(6.6.15)

Next we observe that when $$\tfrac{2} {3} \leq \xi \leq \tfrac{4} {3}$$, only the terms with k = 0 and k = −2 survive in the identity in (6.6.14). This is because when k = −1, $$\xi +k =\xi -1 \in \big [-\tfrac{1} {3}, \tfrac{1} {3}\big]$$ and this interval has null intersection with the support of $$\widehat{\varphi }$$. Therefore, (6.6.14) reduces to

hence

(6.6.16)

Replacing first ν by ν − 1 and then $$\frac{\xi }{2}$$ by ξ in (6.6.16), we obtain

(6.6.17)

Now consider the 2 × 2 system of equations given by (6.6.15) and (6.6.17) with unknown $$\widehat{f}(2^{\nu }(\xi -1))$$ and $$\widehat{f}(2^{\nu }\xi )$$. The determinant of the system is

Therefore, the system has the unique solution

which is valid for all $$\nu \in \mathbf{Z}$$ and all $$\xi \in [\tfrac{1} {3}, \tfrac{2} {3}]$$. We conclude that $$\widehat{f}(\xi ) = 0$$ for all $$\xi \in \mathbf{R}$$ and thus f = 0. This proves the completeness of the system $$\{\varphi _{\nu,k}\}$$. We conclude that the function $$\varphi$$ is a wavelet. □ 

6.6.4 Sampling

Next we discuss how one can recover a band-limited function by its values at a countable number of points.

Definition 6.6.8.

An integrable function on R n is called band limited if its Fourier transform has compact support.

For every band-limited function there is a B > 0 such that its Fourier transform is supported in the cube [−B, B] n . In such a case we say that the function is band limited on the cube [−B, B] n .

It is an interesting observation that such functions are completely determined by their values at the points x = k∕2B, where k ∈ Z n . We have the following result.

Theorem 6.6.9.

(a) Let f in L ¹ ( R n ) be band limited on the cube [−B,B] n . Then f can be sampled by its values at the points x = k∕2B, where k ∈ Z n . In particular, we have

(6.6.18)

for almost all x ∈ R n .

(b) Suppose that f is band-limited on the cube [−B′,B′] n where 0 < B′ < B. Then f can be sampled by its values at the points x = k∕2B, k ∈ Z n as follows

(6.6.19)

for some Schwartz function Φ that depends on B,B′.

Proof.

Since the function $$\widehat{f}$$ is supported in [−B, B] n , we use Exercise 6.6.2 to obtain

Inserting this identity in the inversion formula

which holds for almost all x ∈ R n since $$\widehat{f}$$ is continuous and therefore integrable over [−B, B] n , we obtain

(6.6.20)

(6.6.21)

This is exactly (6.6.18) when we change k to − k and thus part (a) is proved. For part (b) we argue similarly, except that we replace $$\chi _{[-B,B]^{n}}$$ by $$\widehat{\varPhi }$$, where $$\widehat{\varPhi }$$ is smooth, equal to 1 on [−B′, B′] n and vanishes outside [−B, B] n . Then we can insert the function $$\widehat{\varPhi }(\xi )$$ in (6.6.20) and instead of (6.6.21) we obtain the expression on the right in (6.6.19). □ 

Remark 6.6.10.

Identity (6.6.18) holds for any B″ > B. In particular, we have

for all x ∈ R n whenever f is band-limited in [−B, B] n . In particular, band-limited functions in [−B, B] n can be sampled by their values at the points k∕2B″ for any B″ ≥ B.

However, band-limited functions in [−B, B] n cannot be sampled by the points k∕2B′ for any B′ < B, as the following example indicates.

Example 6.6.11.

For 0 < B′ < B, let f(x) = g(x)sin(2π B′x), where $$\widehat{g}$$ is supported in the interval [−(B − B′), B − B′]. Then f is band limited in [−B, B], but it cannot be sampled by its values at the points k∕2B′, since it vanishes at these points and f is not identically zero if g is not the zero function.

Next, we give a couple of results that relate the L p norm of a given function with the ℓ p norm (or quasi-norm) of its sampled values.

Theorem 6.6.12.

Let f be a tempered ¹ function whose Fourier transform is supported in the closed ball $$\overline{B(0,t)}$$ for some 0 < t < ∞. Assume that f lies in $$L^{p}(\mathbf{R}^{n})$$ for some 0 < p ≤∞. Then there is a constant C(n,p) such that

Proof.

The proof is based on the following fact, whose proof can be found in [131] (Lemma 2.2.3). Let 0 < r < ∞. Then there exists a constant C ₂ = C ₂(n, r) such that for all t > 0 and for all $$\mathcal{C}^{1}$$ functions u on R n whose distributional Fourier transform is supported in the ball | ξ | ≤ t we have

(6.6.22)

where M denotes the Hardy–Littlewood maximal operator.

Notice that f is a $$\mathcal{C}^{\infty }$$ function since its Fourier transform is compactly supported. Assuming (6.6.22), for each k ∈ Z n and x ∈ [0, 1] n we use the mean value theorem to obtain

We raise this inequality to the power p, we integrate over the cube [0, 1] n , we sum over k ∈ Z n , and then we take the 1∕p power. Let $$c_{p} =\max (1,2^{1/p-1})$$ and $$c(n,r,t) = \sqrt{n}\,t(1 + t\sqrt{n})^{n/r}$$. The sum over k and the integral over [0, 1] n yield an integral over R n and thus we obtain

where the last step uses (6.6.22). We now select r = p∕2 if p < ∞ and r to be any number if p = ∞. The required inequality follows from the boundedness of the Hardy-Littlewood maximal operator on L ² if p < ∞ or on $$L^{\infty }$$ if p = ∞. □ 

The next theorem could be considered a partial converse of Theorem 6.6.13

Theorem 6.6.13.

Suppose that an integrable function f has Fourier transform supported in the cube $$[-(\frac{1} {2}-\varepsilon ), \frac{1} {2}-\varepsilon ]^{n}$$ for some $$0 <\varepsilon < 1/2$$ . Furthermore, suppose that the sequence of coefficients $$\{f(k)\}_{k\in \mathbf{Z}^{n}}$$ lies in ℓ p ( Z n ) for some 0 < p ≤∞. Then f lies in $$L^{p}(\mathbf{R}^{n})$$ and the following estimate is valid

(6.6.23)

Proof.

We fix a smooth function $$\widehat{\varPhi }$$ supported in $$[-\frac{1} {2}, \frac{1} {2}]^{n}$$ and equal to 1 on the smaller cube $$[-(\frac{1} {2}-\varepsilon ), \frac{1} {2}-\varepsilon ]^{n}$$. Then we may write f = f ∗Φ, since $$\widehat{\varPhi }$$ is equal to one on the support of $$\widehat{f}$$. Writing $$\widehat{f}$$ in terms of its Fourier series we have

(6.6.24)

Since f is integrable, $$\widehat{f}$$ is continuous and thus integrable over $$[-\frac{1} {2}, \frac{1} {2}]^{n}$$. By Fourier inversion we have

(6.6.25)

for almost all x ∈ R n . Inserting (6.6.25) in (6.6.24) we obtain

This identity combined with the rapid decay of Φ yields (6.6.23) as follows. For 0 < p ≤ 1 we have

while for 1 < p ≤ ∞, setting $$Q = [-\frac{1} {2}, \frac{1} {2}]^{n}$$ we write:

The preceding expression can be viewed as the ℓ p norm of the discrete convolution of the sequences $$\{f(k)\}_{k}$$ and $$\frac{1} {(\sqrt{n}+\vert k\vert )^{N}}$$ and thus it is bounded by a constant multiple of $$\big\|\{f(k)\}_{k}\big\|_{\ell^{p}(\mathbf{Z}^{n})}$$, since the sequence $$\frac{1} {(\sqrt{n}+\vert k\vert )^{N}}$$ is in ℓ ¹(Z n ) if N is large enough. This completes the proof. □ 

Exercise 6.6.6 gives examples of functions for which Theorem 6.6.13 fails if ɛ = 0.

Exercises

6.6.1. (a) Let $$A = [-1,-\tfrac{1} {2})\bigcup [\tfrac{1} {2},1)$$. Show that the family $$\{e^{2\pi imx}\}_{m\in \mathbf{Z}}$$ is an orthonormal basis of L ²(A).

(b) Obtain the same conclusion for the family $$\{e^{2\pi im\cdot x}\}_{m\in \mathbf{Z}^{n}}$$ in L ²(A n ).

[Hint: To show completeness, given f ∈ L ²(A), define h on [0, 1] by setting h(x) = f(x − 1) for $$x \in [0, \frac{1} {2})$$ and h(x) = f(x) for $$x \in [\frac{1} {2},1)$$. Observe that $$\widehat{h}(m) =\widehat{ f}(m)$$ for all $$m \in \mathbf{Z}$$ and expand h in Fourier series.]

6.6.2. Let g be an integrable function on R n .

(a) Suppose that g is supported in [−b, b] n for some b > 0 and that the sequence $$\{\widehat{g}(k/2b)\}_{k\in \mathbf{Z}^{n}}$$ lies in $$\ell^{2}(\mathbf{Z}^{n})$$. Show that

where the series converges in L ²(R n ) and deduce that g is in L ²(R n ).

(b) Suppose that g is supported in [0, b] n for some b > 0 and that the sequence $$\{\widehat{g}(k/b)\}_{k\in \mathbf{Z}^{n}}$$ lies in $$\ell^{2}(\mathbf{Z}^{n})$$. Show that

where the series converges in L ²(R n ) and deduce that g is in L ²(R n ).

(c) When n = 1, obtain the same as the conclusion in part (b) for $$x \in [-b,-\tfrac{b} {2})\bigcup$$ $$[\tfrac{b} {2},b)$$, provided g is supported in this set.

[Hint: Part (c): Use the result in Exercise 6.6.1.]

6.6.3. Show that the sequence of functions

is orthonormal in L ²(R n ).

[Hint: Interpret the functions H k as the Fourier transforms of known functions.]

6.6.4. Prove the following spherical multidimensional version of Theorem 6.6.9. Suppose that $$\widehat{f}$$ is supported in the ball | ξ | ≤ R. Show that

where J a is the Bessel function of order a.

6.6.5. Let $$\{a_{k}\}_{k\in \mathbf{Z}^{n}}$$ be in ℓ p for some $$1 < p < \infty $$. Show that the partial sums

converge in $$\mathcal{S}'(\mathbf{R}^{n})$$ as $$N \rightarrow \infty $$ to an $$L^{p}$$ function on $$\mathbf{R}^{n}$$ whose Fourier transform is supported in [−B, B] n . Here k = (k ₁, …, k n ). Moreover, the $$L^{p}$$ norm of A is controlled by a constant multiple of the ℓ p norm of $$\{a_{k}\}_{k}$$.

6.6.6. Consider the function $$\prod _{j=1}^{n}\sin (\pi x_{j})/(\pi x_{j})$$ on R n to show that Theorem 6.6.13 fails when $$\varepsilon = 0$$ and p ≤ 1. When $$1 < p \leq \infty $$ consider the function $$x_{1} +\prod _{ j=1}^{n}\sin (\pi x_{j})/(\pi x_{j})$$.

6.6.7. (a) Let ψ(x) be a nonzero continuous integrable function on R that satisfies $$\int _{\mathbf{R}}\psi (x)\,dx = 0$$ and

Define the wavelet transform of f in $$L^{2}(\mathbf{R})$$ by setting

when a ≠ 0 and W(f; 0, b) = 0. Show that for any $$f \in L^{2}(\mathbf{R})$$ the following inversion formula holds:

(b) State and prove an analogous wavelet transform inversion property on R n .

[Hint: Apply Theorem 2.​2.​14 (5) in the b-integral and use Fourier inversion.]

6.6.8. (P. Casazza) On R n let e j be the vector whose coordinates are zero everywhere except for the jth entry, which is 1. Set $$q_{j} = e_{j} - \frac{1} {n}\sum _{k=1}^{n}e_{ k}$$ for 1 ≤ j ≤ n and also $$q_{n+1} = \frac{1} {\sqrt{n}}\sum _{k=1}^{n}e_{k}$$. Prove that

for all x ∈ R n . This provides an example of a tight frame on R n .

HISTORICAL NOTES

An early account of square functions in the context of Fourier series appears in the work of Kolmogorov [196], who proved the almost everywhere convergence of lacunary partial sums of Fourier series of periodic square-integrable functions. This result was systematically studied and extended to L p functions, 1 < p < ∞, by Littlewood and Paley [227], [228], [229] using complex-analysis techniques. The real-variable treatment of the Littlewood and Paley theorem was pioneered by Stein [334] and allowed the higher-dimensional extension of the theory. The use of vector-valued inequalities in the proof of Theorem 6.1.2 is contained in Benedek, Calderón, and Panzone [22]. A Littlewood–Paley theorem for lacunary sectors in R ² was obtained by Nagel, Stein, and Wainger [264].

An interesting Littlewood–Paley estimate holds for 2 ≤ p < ∞: There exists a constant C p such that for all families of disjoint open intervals I j in $$\mathbf{R}$$ the estimate $$\|(\sum _{j}\vert (\widehat{f}\,\chi _{I_{j}})\,^{\vee }\,\vert ^{2})^{\frac{1} {2} }\|_{L^{p}} \leq C_{p}\|f\|_{L^{p}}$$ holds for all functions $$f \in L^{p}(\mathbf{R})$$. This was proved by Rubio de Francia [301], but the special case in which $$I_{j} = (j,j\! +\! 1)$$ was previously obtained by Carleson [55]. An alternative proof of Rubio de Francia’s theorem was obtained by Bourgain [34]. A higher-dimensional analogue of this estimate for arbitrary disjoint open rectangles in R n with sides parallel to the axes was obtained by Journé [181]. Easier proofs of the higher-dimensional result were subsequently obtained by Sjölin [326], Soria [329], and Sato [311].

Part (a) of Theorem 6.2.7 is due to Mihlin [254] and the generalization in part (b) to Hörmander [159]. Theorem 6.2.2 can be found in Marcinkiewicz’s article [241] in the context of one-dimensional Fourier series. Calderón and Torchinsky [45] have improved Theorem 6.2.7 in the following way: if for a suitable smooth bump η supported in an annulus the functions $$m(2^{k}\xi )\eta (\xi )$$ lie in the Sobolev space $$L_{\gamma }^{r}$$ uniformly in $$k \in \mathbf{Z}$$, where $$\gamma > n(\frac{1} {p} -\frac{1} {2})$$, 1 < p < 2, $$\frac{1} {r} = \frac{1} {p} -\frac{1} {2}$$, then m lies in $$\mathcal{M}_{p}(\mathbf{R}^{n})$$. The power 6 in estimate (6.2.3) that appears in the statement of Theorem 6.2.2 is not optimal. Tao and Wright [357] proved that in dimension 1, the best power of (p − 1)⁻¹ in this theorem is $$\frac{3} {2}$$ as p → 1. An improvement of the Marcinkiewicz multiplier theorem in one dimension was obtained by Coifman, Rubio de Francia, and Semmes [69]. Weighted norm estimates for Hörmander–Mihlin multipliers were obtained by Kurtz and Wheeden [209] and for Marcinkiwiecz multipliers by Kurtz [208]. Heo, Nazarov, and Seeger [150] have obtained a very elegant characterization of radial L p multipliers in large dimensions; precisely, they showed that for dimensions n ≥ 4 and $$1 < p < \frac{2n-2} {n+1}$$, a radial function m on R n is an L p Fourier multiplier if and only if there exists a nonzero Schwartz function η such that $$\sup _{t>0}t^{n/p}\|(m(\,\cdot \,)\eta (t\,\cdot ))\,^{\vee }\,\|_{L^{p}} < \infty $$. This characterization builds on and extends a previously obtained simple characterization by Garrigós and Seeger [124] of radial multipliers on the invariant subspace of radial L p functions when $$1 < p < \frac{2n} {n+1}$$.

The method of proof of Theorem 6.3.4 is adapted from Duoandikoetxea and Rubio de Francia [102]. The method in this article is rather general and can be used to obtain L p boundedness for a variety of rough singular integrals. A version of Theorem 6.3.6 was used by Christ [59] to obtain L p smoothing estimates for Cantor–Lebesgue measures. When p = q ≠ 2, Theorem 6.3.6 is false in general, but it is true for all r satisfying $$\vert \frac{1} {r} -\frac{1} {2}\vert < \vert \frac{1} {p} -\frac{1} {2}\vert $$ under the additional assumption that the m j ’s are Lipschitz functions uniformly at all scales. This result was independently obtained by Carbery [52] and Seeger [316]. Miyachi [255] has obtained a complete characterization of the indices a, b > 0 such that the functions $$\vert x\vert ^{-b}e^{i\vert x\vert ^{a} }\psi (x)$$ are L p Fourier multipliers; here ψ is a smooth function that is equal to 1 near infinity and vanishes near zero.

The probabilistic notions of conditional expectations and martingales have a strong connection with the Littlewood–Paley theory discussed in this chapter. For the purposes of this exposition we considered only the case of the sequence of σ-algebras generated by the dyadic cubes of side length 2−k in R n . The L p boundedness of the maximal conditional expectation (Doob [97]) is analogous to the L p boundedness of the dyadic maximal function; likewise with the corresponding weak type (1, 1) estimate. The L p boundedness of the dyadic martingale square function was obtained by Burkholder [39] and is analogous to Theorem 6.1.2. Moreover, the estimate $$\big\|\sup _{k}\vert E_{k}(f)\vert \big\|_{L^{p}} \approx \big\| S(f)\big\|_{L^{p}}$$, 0 < p < ∞, obtained by Burkholder and Gundy [40] and also by Davis [90] is analogous to the square-function characterization of the Hardy space H p norm. For an exposition on the different and unifying aspects of Littlewood–Paley theory we refer to Stein [337]. The proof of Theorem 6.4.8, which quantitatively expresses the almost orthogonality of the Littlewood–Paley and the dyadic martingale difference operators, is taken from Grafakos and Kalton [133].

The use of quadratic expressions in the study of certain maximal operators has a long history. We refer to the article of Stein [340] for a historical survey. Theorem 6.5.1 was first proved by Stein [339]. The proof in the text is taken from an article of Rubio de Francia [302]. Another proof when n ≥ 3 is due to Cowling and Mauceri [76]. The more difficult case n = 2 was settled by Bourgain [36] about 10 years later. Alternative proofs when n = 2 were given by Mockenhaupt, Seeger, and Sogge [256] as well as Schlag [313]. The boundedness of maximal operators associated to more general smooth measures on compact surfaces of finite type were investigated by Iosevich and Sawyer [173]. The powerful machinery of Fourier integral operators was used by Sogge [328] to obtain the boundedness of spherical maximal operators on compact manifolds without boundary and positive injectivity radius; a simple proof for the boundedness of the spherical maximal function on the sphere was given by Nguyen [269]. Weighted norm inequalities for the spherical maximal operator were obtained by Duoandikoetxea and Vega [103]. The discrete spherical maximal function was studied by Magyar, Stein, and Wainger [237].

Much of the theory of square functions and the ideas associated with them has analogues in the dyadic setting. A dyadic analogue of the theory discussed here can be obtained. For an introduction to the area of dyadic harmonic analysis, we refer to Pereyra [276].

The idea of expressing (or reproducing) a signal as a weighted average of translations and dilations of a single function appeared in early work of Calderón [42]. This idea is in some sense a forerunner of wavelets. An early example of a wavelet was constructed by Strömberg [352] in his search for unconditional bases for Hardy spaces. Another example of a wavelet basis was obtained by Meyer [249]. The construction of an orthonormal wavelet presented in Theorem 6.6.7 is in Lemarié and Meyer [216]. A compactly supported wavelet was constructed by Daubechies [88]. Mallat [238] introduced the notion of multiresolution analysis, which led to a systematic production of wavelets. Theorem 6.6.9 is Shannon’s [319] version of Nyquist’s theorem [270] and is referred to as the Nyquist-Shannon sampling theorem. It is a fundamental result in telecommunications and signal processing, since it describes how to reconstruct a signal that contains no frequencies higher than B Hertz in terms of its values at a sequence of points spaced 1∕(2B) seconds apart.

The area of wavelets has taken off significantly since its inception, spurred by these early results. A general theory of wavelets and its use in Fourier analysis was carefully developed in the two-volume monograph of Meyer [250], [251] and its successor Meyer and Coifman [253]. For further study and a deeper account of developments on the subject the reader may consult the books of Daubechies [89], Chui [64], Wickerhauser [374], Kaiser [184], Benedetto and Frazier [23], Hérnandez and Weiss [151], Wojtaszczyk [379], Mallat [239], Meyer [252], Frazier [120], Gröchenig [140], and the references therein. Theorems 6.6.12 and 6.6.13 first appeared in a combined form in the work of Plancherel and Pólya [285] for restrictions of entire functions of exponential type on the real line.

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Footnotes

1

A function is called tempered if there are constants C,M such that $$\vert f(x)\vert \leq C\,(1 + \vert x\vert )^{M}$$ for all x ∈ R n . Tempered functions are tempered distributions.

© Springer Science+Business Media New York 2014

Loukas GrafakosClassical Fourier AnalysisGraduate Texts in Mathematics24910.1007/978-1-4939-1194-3_7

7. Weighted Inequalities

Loukas Grafakos1 

(1)

Department of Mathematics, University of Missouri, Columbia, MO, USA

Weighted inequalities arise naturally in Fourier analysis, but their use is best justified by the variety of applications in which they appear. For example, the theory of weights plays an important role in the study of boundary value problems for Laplace’s equation on Lipschitz domains. Other applications of weighted inequalities include extrapolation theory, vector-valued inequalities, and estimates for certain classes of nonlinear partial differential equations.

The theory of weighted inequalities is a natural development of the principles and methods we have acquainted ourselves with in earlier chapters. Although a variety of ideas related to weighted inequalities appeared almost simultaneously with the birth of singular integrals, it was only in the 1970s that a better understanding of the subject was obtained. This was spurred by Muckenhoupt’s characterization of positive functions w for which the Hardy–Littlewood maximal operator M maps L p (R n , w(x) d x) to itself. This characterization led to the introduction of the class A p and the development of weighted inequalities. We pursue exactly this approach in the next section to motivate the introduction of the A p classes.

7.1 The A p Condition

A weight is a nonnegative locally integrable function on R n that takes values in (0, ∞) almost everywhere. Therefore, weights are allowed to be zero or infinite only on a set of Lebesgue measure zero. Hence, if w is a weight and 1∕w is locally integrable, then 1∕w is also a weight.

Given a weight w and a measurable set E, we use the notation

to denote the w-measure of the set E. Since weights are locally integrable functions, w(E) < ∞ for all sets E contained in some ball. The weighted L p spaces are denoted by L p (R n , w) or simply L p (w). Recall the uncentered Hardy–Littlewood maximal operators on R n over balls

and over cubes

where the suprema are taken over all balls B and cubes Q (with sides parallel to the axes) that contain the given point x. A classical result (Theorem 2.​1.​6) states that for all 1 < p < ∞ there is a constant C p (n) > 0 such that

(7.1.1)

for all functions f ∈ L p (R n ). We are concerned with the situation in which the measure d x in (7.1.1) is replaced by w(x) d x for some weight w(x).

7.1.1 Motivation for the A p Condition

The question we raise is whether there is a characterization of all weights w(x) such that the strong type (p, p) inequality

(7.1.2)

is valid for all f ∈ L p (w).

Suppose that (7.1.2) is valid for some weight w and all f ∈ L p (w) for some 1 < p < ∞. Apply (7.1.2) to the function f χ B supported in a ball B and use that $$\mathop{\mbox{ Avg}}_{B}\vert f\vert \leq M(f\chi _{B})(x)$$ for all x ∈ B to obtain

(7.1.3)

It follows that

(7.1.4)

for all balls B and all functions f. At this point, it is tempting to choose a function such that the two integrands are equal. We do so by setting $$f = w^{-p'/p}$$, which gives $$f^{p}w = w^{-p'/p}$$. Under the assumption that $$\inf _{B}w > 0$$ for all balls B, it would follow from (7.1.4) that

(7.1.5)

If $$\inf _{B}w = 0$$ for some balls B, we take $$f = (w+\varepsilon )^{-p'/p}$$ to obtain

(7.1.6)

for all $$\varepsilon > 0$$. Replacing w(x) d x by $$(w(x)+\varepsilon )\,dx$$ in the last integral in (7.1.6) we obtain a smaller expression, which is also bounded by C p p . Since $$-p'/p = -p' + 1$$, (7.1.6) implies that

(7.1.7)

from which we can still deduce (7.1.5) via the Lebesgue monotone convergence theorem by letting $$\varepsilon \rightarrow 0$$. We have now obtained that every weight w that satisfies (7.1.2) must also satisfy the rather strange-looking condition (7.1.5), which we refer to in the sequel as the A p condition. It is a remarkable fact, to be proved in this chapter, that the implication obtained can be reversed, that is, (7.1.2) is a consequence of (7.1.5). This is the first significant achievement of the theory of weights [i.e., a characterization of all functions w for which (7.1.2) holds]. This characterization is based on some deep principles discussed in the next section and provides a solid motivation for the introduction and careful examination of condition (7.1.5).

Before we study the converse statements, we consider the case p = 1. Assume that for some weight w the weak type (1, 1) inequality

(7.1.8)

holds for all functions f ∈ L ¹(R n ). Since $$M(f)(x) \geq \mathop{\mbox{ Avg}}_{B}\vert f\vert $$ for all x ∈ B, it follows from (7.1.8) that for all $$\alpha <\mathop{ \mbox{ Avg}}_{B}\vert f\vert $$ we have

(7.1.9)

Taking f χ B instead of f in (7.1.9), we deduce that

(7.1.10)

for all functions f and balls B. Taking f = χ S , we obtain

(7.1.11)

where S is any measurable subset of the ball B.

Recall that the essential infimum of a function w over a set E is defined as

Then for every $$a >\mathop{ \mbox{ ess.inf}}_{B}(w)$$ there exists a subset S a of B with positive measure such that w(x) < a for all x ∈ S a . Applying (7.1.11) to the set S a , we obtain

(7.1.12)

which implies

(7.1.13)

It remains to understand what condition (7.1.13) really means. For every ball B, there exists a null set N(B) such that (7.1.13) holds for all x in B ∖ N(B). Let N be the union of all the null sets N(B) for all balls B with centers in Q n and rational radii. Then N is a null set and for every x in B ∖ N, (7.1.13) holds for all balls B with centers in Q n and rational radii. By density, (7.1.13) must also hold for all balls B that contain a fixed x in R n ∖ N. It follows that for x ∈ R n ∖ N we have

(7.1.14)

Therefore, assuming (7.1.8), we have arrived at the condition

(7.1.15)

where C ₁ is the same constant as in (7.1.13).

We later see that this deduction can be reversed and we can obtain (7.1.8) as a consequence of (7.1.15). This motivates a careful study of condition (7.1.15), which we refer to as the A ₁ condition. Since in all the previous arguments we could have replaced balls with cubes, we give the following definitions in terms of cubes.

Definition 7.1.1.

A function w(x) ≥ 0 is called an A ₁ weight if

(7.1.16)

for some constant C ₁. If w is an A ₁ weight, then the (finite) quantity

(7.1.17)

is called the A ₁ Muckenhoupt characteristic constant of w, or simply the A ₁ characteristic constant of w. Note that A ₁ weights w satisfy

(7.1.18)

for all cubes Q in R n .

Remark 7.1.2.

We also define

(7.1.19)

Using (7.1.13), we see that the smallest constant C ₁ that appears in (7.1.16) is equal to the A ₁ characteristic constant of w as defined in (7.1.19). This is also equal to the smallest constant that appears in (7.1.13). All these constants are bounded above and below by dimensional multiples of $$[w]_{A_{1}}$$.

We now recall condition (7.1.5), which motivates the following definition of A p weights for 1 < p < ∞.

Definition 7.1.3.

Let 1 < p < ∞. A weight w is said to be of class A p if

(7.1.20)

The expression in (7.1.20) is called the A p Muckenhoupt characteristic constant of w (or simply the A p characteristic constant of w) and is denoted by $$[w]_{A_{p}}$$.

Remark 7.1.4.

Note that Definitions 7.1.1 and 7.1.3 could have been given with the set of all cubes in R n replaced by the set of all balls in R n . Defining $$[w]_{A_{p}}^{\text{balls}}$$ as in (7.1.20) except that cubes are replaced by balls, we see that

(7.1.21)

7.1.2 Properties of A p Weights

It is straightforward that translations, isotropic dilations, and scalar multiples of A p weights are also A p weights with the same A p characteristic. We summarize some basic properties of A p weights in the following proposition.

Proposition 7.1.5.

Let w ∈ A p for some 1 ≤ p < ∞. Then

(1)

$$[\delta ^{\lambda }(w)]_{A_{p}} = [w]_{A_{p}}$$ , where $$\delta ^{\lambda }(w)(x) = w(\lambda x_{1},\ldots,\lambda x_{n})$$ .

(2)

$$[\tau ^{z}(w)]_{A_{p}} = [w]_{A_{p}}$$ , where $$\tau ^{z}(w)(x) = w(x - z)$$ , z ∈ R n .

(3)

$$[\lambda w]_{A_{p}} = [w]_{A_{p}}$$ for all λ > 0.

(4)

When 1 < p < ∞, the function $$w^{- \frac{1} {p-1} }$$ is in A p′ with characteristic constant

Therefore, w ∈ A ₂ if and only if w ⁻¹ ∈ A ₂ and both weights have the same A ₂ characteristic constant.

(5)

$$[w]_{A_{p}} \geq 1$$ for all w ∈ A p . Equality holds if and only if w is a constant.

(6)

The classes A p are increasing as p increases; precisely, for 1 ≤ p < q < ∞ we have

(7)

$$\lim \limits _{q\rightarrow 1+}[w]_{A_{q}} = [w]_{A_{1}}$$ if w ∈ A ₁ .

(8)

The following is an equivalent characterization of the A p characteristic constant of w:

(9)

The measure w(x) dx is doubling: precisely, for all λ > 1 and all cubes Q we have

(λQ denotes the cube with the same center as Q and side length λ times the side length of Q.)

Proof.

The simple proofs of (1), (2), and (3) are left as an exercise. Property (4) is also easy to check and plays the role of duality in this context. To prove (5) we use Hölder’s inequality with exponents p and p′ to obtain

with equality holding only when $$w(x)^{\frac{1} {p} } = c\,w(x)^{-\frac{1} {p} }$$ for some c > 0 (i.e., when w is a constant). To prove (6), observe that $$0 < q' - 1 < p' - 1 \leq \infty $$ and that the statement

is equivalent to the fact

Property (7) is a consequence of part (a) of Exercise 1.1.3.

To prove (8), apply Hölder’s inequality with exponents p and p′ to get

This argument proves the inequality ≥ in (8) when p > 1. In the case p = 1 the obvious modification yields the same inequality. The reverse inequality follows by taking $$f = (w+\varepsilon )^{-p'/p}$$ as in (7.1.6) and letting $$\varepsilon \rightarrow 0$$.

Applying (8) to the function f = χ Q and putting λ Q in the place of Q in (8), we obtain

which says that w(x) d x is a doubling measure. This proves (9). □ 

Example 7.1.6.

A positive measure d μ is called doubling if for some C < ∞,

(7.1.22)

for all balls B. We show that the measures | x |  a  d x are doubling when a > −n. We divide all balls B(x ₀, R) in R n into two categories: balls of type I that satisfy | x ₀ | ≥ 3R and type II that satisfy | x ₀ |  < 3R. For balls of type I we observe that

Since | x ₀ | ≥ 3R, we have $$\vert x_{0}\vert + 2R \leq 4(\vert x_{0}\vert - R)$$ and $$\vert x_{0}\vert - 2R \geq \frac{1} {4}(\vert x_{0}\vert + R)$$, from which (7.1.22) follows with C = 23n 4 | a | .

For balls of type II, we have | x ₀ | ≤ 3R and we note two things: first

and second, since | x |  a is radially decreasing for a < 0 and radially increasing for a ≥ 0, we have

For $$x \in B(3R \frac{x_{0}} {\vert x_{0}\vert },R)$$ we must have | x | ≥ 2R, and hence both integrals on the right are at least a multiple of R n+a . This establishes (7.1.22) for balls of type II.

Example 7.1.7.

We investigate for which real numbers a the power function | x |  a is an A p weight on R n . For 1 < p < ∞, we examine for which a the following expression is finite:

(7.1.23)

As in the previous example we split the balls in R n into those of type I and those of type II. If B = B(x ₀, R) is of type I, then for x satisfying | x − x ₀ | ≤ R we must have

thus the expression inside the supremum in (7.1.23) is comparable to

If B(x ₀, R) is a ball of type II, then B(0, 5R) has size comparable to B(x ₀, R) and contains it. Since the measure | x |  a  d x is doubling, the integrals of the function | x |  a over B(x ₀, R) and over B(0, 5R) are comparable. It suffices therefore to estimate the expression inside the supremum in (7.1.23), in which we have replaced B(x ₀, R) by B(0, 5R). But this is

which is seen easily to be finite and independent of R exactly when $$-n < a < n\frac{p} {p'}$$. We conclude that | x |  a is an A p weight, 1 < p < ∞, if and only if $$-n < a < n(p - 1)$$.

The previous proof can be suitably modified to include the case p = 1. In this case we obtain that | x |  a is an A ₁ weight if and only if − n < a ≤ 0. As we have seen, the measure | x |  a  d x is doubling on the larger range − n < a < ∞. Thus for a > n(p − 1), the function | x |  a provides an example of a doubling measure that is not in A p .

Example 7.1.8.

On R n the function

is an A ₁ weight. Indeed, to check condition (7.1.19) it suffices to consider balls of type I and type II as defined in Example 7.1.6. In either case the required estimate follows easily.

We now return to a point alluded to earlier, that the A p condition implies the boundedness of the Hardy–Littlewood maximal function M on the space L p (w). To this end we introduce four maximal functions acting on functions f that are locally integrable with respect to w:

where the supremum is taken over open balls B that contain the point x and

where Q is an open cube containing the point x, and

where $$Q(x,\delta ) =\prod _{ j=1}^{n}(x_{j}-\delta,x_{j}+\delta )$$ is a cube of side length 2δ centered at $$x = (x_{1},\ldots,x_{n})$$. When w = 1, these maximal functions reduce to the standard ones M(f), $$\mathcal{M}(f)$$, $$M_{c}(f)$$, and $$\mathcal{M}_{c}(f)$$, the uncentered and centered Hardy–Littlewood maximal functions with respect to balls and cubes, respectively.

Theorem 7.1.9.

(a) Let w ∈ A ₁ . Then we have

(7.1.24)

(b) Let w ∈ A p ( R n ) for some 1 < p < ∞. Then there is a constant C n,p such that

(7.1.25)

Since the operators $$\mathcal{M}_{c}$$ , M c , $$\mathcal{M}$$ , and M are pointwise comparable, a similar conclusions hold for the other three as well.

Proof.

(a) Since d μ = w d x is a doubling measure and $$d\mu (3Q) \leq 3^{n}[w]_{A_{1}}\mu (Q)$$, using Proposition 7.1.5 (9) and Exercise 2.1.1 we obtain that M c w maps L ¹(w) to L 1, ∞ (w) with norm at most $$3^{n}[w]_{A_{1}}$$. This proves (7.1.24).

(b) Fix a weight w in A p and let $$\sigma = w^{- \frac{1} {p-1} }$$ be its dual weight. Fix an open cube Q = Q(x ₀, r) in R n with center x ₀ and side length 2r and write

(7.1.26)

For any x ∈ Q, consider the cube Q(x, 2r). Then $$Q \subseteq Q(x,2r) \subseteq 3Q = Q(x_{0},3r)$$ and thus

for any x ∈ Q. Inserting this expression in (7.1.26), we obtain

(7.1.27)

Since one may easily verify that

it follows that

since x ₀ is the center of Q. Hence, we have

Applying L p (w) norms, we deduce

and conclusion (7.1.25) follows, provided we show that

(7.1.28)

for any 1 < q < ∞ and any weight w.

We obtain this estimate by interpolation. Obviously (7.1.28) is valid when q = ∞ with C(∞, n) = 1. If we prove that

(7.1.29)

then (7.1.28) will follow from Theorem 1.​3.​2.​

To prove (7.1.29) we fix f ∈ L ¹(R n , w d x). We first show that the set

is open. For any r > 0, let Q(x, r) denote an open cube of side length 2r with center x ∈ R n . If we show that for any r > 0 and x ∈ R n the function

(7.1.30)

is continuous, then $$\mathcal{M}_{c}^{w}(f)$$ is the supremum of continuous functions; hence it is lower semicontinuous and thus the set E λ is open. But this is straightforward. If x n  → x ₀, then w(Q(x n , r)) → w(Q(x ₀, r)) and also $$\int _{Q(x_{n},r)}\vert f\vert \,w\,dy \rightarrow \int _{Q(x_{0},r)}\vert f\vert \,w\,dy$$ by the Lebesgue dominated convergence theorem. Since w(Q(x ₀, r)) ≠ 0, it follows that the function in (7.1.30) is continuous.

Given K a compact subset of E λ , for any x ∈ K select an open cube Q x centered at x such that

Applying Lemma 7.1.10 (proved immediately afterward) we find a subfamily $$\{Q_{x_{j}}\}_{j=1}^{m}$$ of the family of the balls {Q x :  x ∈ K} such that (7.1.31) and (7.1.32) hold. Then

where the last inequality follows by multiplying (7.1.32) by | f | w and integrating over R n . Taking the supremum over all compact subsets K of E λ and using the inner regularity of w d x, which is a consequence of the Lebesgue monotone convergence theorem, we deduce that $$\mathcal{M}_{c}^{w}$$ maps L ¹(w) to L 1, ∞ (w) with constant at most 24 n . Thus (7.1.29) holds with C(1, n) = 24 n . □ 

Lemma 7.1.10.

Let K be a bounded set in R n and for every x ∈ K, let Q x be an open cube with center x and sides parallel to the axes. Then there are an $$m \in \mathbf{Z}^{+} \cup \{\infty \}$$ and a sequence of points $$\{x_{j}\}_{j=1}^{m}$$ in K such that

(7.1.31)

and for almost all y ∈ R n one has

(7.1.32)

Proof.

Let $$s_{0} =\sup \{\ell (Q_{x}):\,\, x \in K\}.$$ If s ₀ = ∞, then there exists x ₁ ∈ K such that $$\ell(Q_{x_{1}}) > 4L$$, where [−L, L] n contains K. Then K is contained in $$Q_{x_{1}}$$ and the statement of the lemma is valid with m = 1.

Suppose now that s ₀ < ∞. Select x ₁ ∈ K such that $$\ell(Q_{x_{1}}) > s_{0}/2$$. Then define

and select x ₂ ∈ K ₁ such that $$\ell(Q_{x_{2}}) > s_{1}/2$$. Next define

and select x ₃ ∈ K ₂ such that $$\ell(Q_{x_{3}}) > s_{2}/2$$. Continue until the first integer m is found such that K m is an empty set. If no such integer exists, continue this process indefinitely and set m = ∞.

We claim that for all i ≠ j we have $$\frac{1} {3}Q_{x_{i}} \cap \frac{1} {3}Q_{x_{j}} = \varnothing $$. Indeed, suppose that i > j. Then $$x_{i} \in K_{i-1} = K\setminus (Q_{x_{1}} \cup \ldots \cup Q_{x_{i-1}})$$; thus x i ∉Q j . Also $$x_{i} \in K_{i-1} \subseteq K_{j-1}$$, which implies that $$\ell(Q_{x_{i}}) \leq s_{j-1} < 2\ell(Q_{x_{j}})$$. If x i ∉Q j and $$\ell(Q_{x_{j}}) > \frac{1} {2}\ell(Q_{x_{i}})$$, it easily follows that $$\frac{1} {3}Q_{x_{i}} \cap \frac{1} {3}Q_{x_{j}} = \varnothing $$.

We now prove (7.1.31). If m < ∞, then K m  = ∅ and therefore $$K \subseteq \bigcup _{j=1}^{m}Q_{x_{j}}$$. If m = ∞, then there is an infinite number of selected cubes $$Q_{x_{j}}$$. Since the cubes $$\frac{1} {3}Q_{x_{j}}$$ are pairwise disjoint and have centers in a bounded set, it must be the case that some subsequence of the sequence of their lengths converges to zero. If there exists a $$y \in K\setminus \bigcup _{j=1}^{\infty }Q_{x_{j}}$$, this y would belong to all K j , $$j = 1,2,\ldots$$, and then s j  ≥ ℓ(Q y ) for all j. Since some subsequence of the s j ’s tends to zero, it would follow that ℓ(Q y ) = 0, which would force the open cube Q y to be the empty set, a contradiction. Thus (7.1.31) holds.

Finally, we show that $$\sum _{j=1}^{m}\chi _{Q_{x_{ j}}}(y) \leq 24^{n}$$ for almost every point y ∈ R n . To prove this we consider the n hyperplanes H i that are parallel to the coordinate hyperplanes and pass through the point y. Then we write R n as a union of n hyperplanes H i of n-dimensional Lebesgue measure zero and 2 n higher-dimensional open “octants” O r , henceforth called orthants. We fix a y ∈ R n and we show that there are only 12 n points x j such that y lies in $$O_{r} \cap Q_{x_{j}}$$ for a given open orthant O r . To prove this assertion, setting $$\vert z\vert _{\ell^{\infty }} =\sup _{1\leq i\leq n}\vert z_{i}\vert $$ for points $$z = (z_{1},\ldots,z_{n})$$ in R n , we pick an $$x_{k_{0}} \in K \cap O_{r}$$ such that $$Q_{x_{k_{ 0}}}$$ contains y and $$\vert x_{k_{0}} - y\vert _{\ell^{\infty }}$$ is the largest possible among all $$\vert x_{j} - y\vert _{\ell^{\infty }}$$. If x j is another point in K ∩ O r such that $$Q_{x_{j}}$$ contains y, then we claim that $$x_{j} \in Q_{x_{k_{ 0}}}$$. Indeed, to show this we notice that for each $$i \in \{ 1,\ldots,n\}$$ we have

where the second equality is due to the fact that $$x_{j},x_{k_{0}}$$ lie in the same orthant and the last inequality in the fact that $$y \in Q_{x_{k_{ 0}}}$$; it follows that x j lies in $$Q_{x_{k_{ 0}}}$$.

We observed previously that i > j implies x i ∉Q j . Since x j lies in $$Q_{x_{k_{ 0}}}$$, one must then have j ≤ k ₀, which implies that $$\frac{1} {2}\ell(Q_{x_{k_{0}}}) <\ell (Q_{x_{j}})$$. Thus all cubes $$Q_{x_{j}}$$ with centers in K ∩ O r that contain the fixed point y have side lengths comparable to that of $$Q_{x_{k_{ 0}}}$$. A simple geometric argument now gives that there are at most finitely many cubes $$Q_{x_{j}}$$ of side length between α and 2α that contain the given point y such that $$\frac{1} {3}Q_{x_{j}}$$ are pairwise disjoint. Indeed, let $$\alpha = \frac{1} {2}\ell(Q_{x_{k_{0}}})$$ and let $$\{Q_{x_{r}}\}_{r\in I}$$ be the cubes with these properties. Then we have

since all the cubes $$Q_{x_{r}}$$ contain the point y and have length at most 2α and they must therefore be contained in a cube of side length 4α centered at y. This observation shows that | I | ≤ 12 n , and since there are 2 n sets O r , we conclude the proof of (7.1.32). □ 

Remark 7.1.11.

Without use of the covering Lemma 7.1.10, (7.1.29) can be proved via the doubling property of w (cf. Exercise 2.1.1(a)), but then the resulting constant C(q, n) would depend on the doubling constant of the measure w d x and thus on $$[w]_{A_{p}}$$; this would yield a worse dependence on $$[w]_{A_{p}}$$ in the constant in (7.1.25).

Exercises

7.1.1. Let k be a nonnegative measurable function such that k, k ⁻¹ are in $$L^{\infty }(\mathbf{R}^{n})$$. Prove that if w is an A p weight for some 1 ≤ p < ∞, then so is k w.

7.1.2. Let w ₁, w ₂ be two A ₁ weights and let 1 < p < ∞. Prove that w ₁ w ₂ 1−p is an A p weight by showing that

7.1.3. Suppose that w ∈ A p for some p ∈ [1, ∞) and 0 < δ < 1. Prove that w δ  ∈ A q , where $$q =\delta p + 1-\delta$$, by showing that

7.1.4. Show that if the A p characteristic constants of a weight w are uniformly bounded for all p > 1, then w ∈ A ₁.

7.1.5. Let $$w_{0} \in A_{p_{0}}$$ and $$w_{1} \in A_{p_{1}}$$ for some 1 ≤ p ₀, p ₁ < ∞. Let 0 ≤ θ ≤ 1 and define

Prove that

thus w is in A p .

7.1.6. ([122]) Fix 1 < p < ∞. A pair of weights (u, w) that satisfies

is said to be of class (A p , A p ). The quantity $$[u,w]_{(A_{p},A_{p})}$$ is called the (A p , A p ) characteristic constant of the pair (u, w).

(a) Suppose that pair of weights (u, w) is of class (A p , A p ). Show that for all nonnegative measurable functions f and all cubes Q′ we have

where $$C_{0} = [u,w]_{(A_{p},A_{p})}$$.

(b) Suppose that a pair of weights (u, w) satisfies the inequality in part (a) for some constant C ₀. Prove that M maps L p (w) to L p, ∞ (u) with norm at most C(n, p)C ₀ 1∕p , where C(n, p) is a fixed constant.

(c) Suppose that for a pair of weights (u, w), M maps L p (w) to L p, ∞ (u). Show that the pair (u, w) is of class (A p , A p ).

[Hint: Part (b): Replacing f by f χ Q in part (a), where Q ⊆ Q′, obtain that

Then use Exercise 5.3.9 to find disjoint cubes Q j such that the set $$E_{\alpha } =\{ x \in \mathbf{R}^{n}:\,\, M_{c}(f)(x) >\alpha \}$$ is contained in the union of 3Q j and $$\frac{\alpha }{4^{n}} < \frac{1} {\vert Q_{j}\vert }\int _{Q_{j}}\vert f(t)\vert \,dt \leq \frac{\alpha } {2^{n}}$$. Then $$u(E_{\alpha }) \leq \sum _{j}u(3Q_{j})$$, and bound each u(3Q j ) by taking Q′ = 3Q j and Q = Q j in the preceding estimate. Part (c): First prove the assertion in part (b) and then derive the inequality in part (a) by adapting the idea in the discussion in the beginning of Subsection 7.1.1.]

7.1.7. ([122]) Let 1 < p < ∞ and let (u, w) be a pair of weights of class (A p , A p ). Show that for any q with p < q < ∞ there is a constant C p, q, n  < ∞ such that for all f ∈ L q (w) we have

[Hint: Use Exercise 7.1.6 and interpolate between L p and L ∞ .]

7.1.8. Let k > 0. For an A ₁ weight w show that $$[\min (w,k)]_{A_{1}} \leq [w]_{A_{1}}$$. If 1 < p < ∞ and w ∈ A p , show that

where c p  = 1 if 1 < p ≤ 2 and $$c_{p} = 2^{p-1}$$ if 2 < p < ∞.

[Hint: Use the inequality $$\frac{1} {\vert Q\vert }\int _{Q}\min (w,k)^{- \frac{1} {p-1} }dx \leq \frac{1} {\vert Q\vert }\int _{Q}w^{- \frac{1} {p-1} }dx + k^{- \frac{1} {p-1} }$$ and also $$\frac{1} {\vert Q\vert }\int _{Q}\min (w,k)dx \leq \min \big\{ k, \frac{1} {\vert Q\vert }\int _{Q}w\,dx\big\}$$.]

7.1.9. Suppose that $$w_{j} \in A_{p_{j}}$$ with 1 ≤ j ≤ m for some $$1 \leq p_{1},\ldots,p_{m} < \infty $$ and let $$0 <\theta _{1},\ldots,\theta _{m} < 1$$ be such that $$\theta _{1} + \cdots +\theta _{m} = 1$$. Show that

[Hint: First note that each weight w j lies in $$A_{\max \{p_{1},\ldots,p_{m}\}}$$ and then apply Hölder’s inequality.]

7.1.10. Let $$w_{1} \in A_{p_{1}}$$ and $$w_{2} \in A_{p_{2}}$$ for some 1 ≤ p ₁, p ₂ < ∞. Prove that

where p = max(p ₁, p ₂).

7.1.11. Show that the function

in Example 7.1.8 is an A ₁ weight on R n .

[Hint: Use $$[u]_{A_{1}}^{\mbox{ balls}}$$ instead of $$[u]_{A_{1}}$$ and consider balls of type I and II as in Example 7.1.7.]

7.1.12. Let 1 < p < ∞ and w ∈ A ₁. Show that the uncentered Hardy-Littlewood maximal function M maps L p, ∞ (w) to itself.

[Hint: Prove first the inequality

and then use the characterization of L p, ∞ given in Exercise 1.1.12.]

7.2 Reverse Hölder Inequality for A p Weights and Consequences

An essential property of A p weights is that they assign to subsets of balls mass proportional to the percentage of the Lebesgue measure of the subset within the ball. The following lemma provides a way to quantify this statement.

Lemma 7.2.1.

Let w ∈ A p for some 1 ≤ p < ∞ and let 0 < α < 1. Then there exists β < 1 such that whenever S is a measurable subset of a cube Q that satisfies |S|≤α|Q|, we have w(S) ≤β w(Q).

Proof.

Taking f = χ A in property (8) of Proposition 7.1.5, we obtain

(7.2.1)

We write S = Q ∖ A to get

(7.2.2)

Given 0 < α < 1, set

(7.2.3)

and use (7.2.2) to obtain the required conclusion. □ 

7.2.1 The Reverse Hölder Property of A p Weights

We are now ready to state and prove one of the main results of the theory of weights, the reverse Hölder inequality for A p weights.

Theorem 7.2.2.

Let w ∈ A p for some 1 ≤ p < ∞. Then there exist constants C and γ > 0 that depend only on the dimension n, on p, and on $$[w]_{A_{p}}$$ such that for every cube Q we have

(7.2.4)

Proof.

Let us fix a cube Q and set

We also fix 0 < α < 1. We define an increasing sequence of scalars

for k ≥ 0 by setting

and for each k ≥ 1 we apply a Calderón–Zygmund decomposition to w at height α k . Precisely, for dyadic subcubes R of Q, we let

(7.2.5)

be the selection criterion. Since Q does not satisfy the selection criterion, it is not selected. We divide the cube Q into a mesh of 2 n subcubes of equal side length, and among these cubes we select those that satisfy (7.2.5). We subdivide each unselected subcube into 2 n cubes of equal side length and we continue in this way indefinitely. We denote by {Q k, j } j the collection of all selected subcubes of Q. We observe that the following properties are satisfied:

(1)

$$\alpha _{k} < \dfrac{1} {\vert Q_{k,j}\vert }\int _{Q_{k,j}}w(t)\,dt \leq 2^{n}\alpha _{ k}$$.

(2)

For almost all x∉U k we have w(x) ≤ α k , where $$U_{k} =\bigcup \limits _{j}Q_{k,j}.$$

(3)

Each Q k+1, j is contained in some Q k, l .

Property (1) is satisfied since the unique dyadic parent of Q k, j was not chosen in the selection procedure. Property (2) follows from the Lebesgue differentiation theorem using the fact that for almost all x∉U k there exists a sequence of unselected cubes of decreasing lengths whose closures’ intersection is the singleton {x}. Property (3) is satisfied since each Q k, j is the maximal subcube of Q satisfying (7.2.5). And since the average of w over Q k+1, j is also bigger than α k , it follows that Q k+1, j must be contained in some maximal cube that possesses this property.

We now compute the portion of Q k, l that is covered by cubes of the form Q k+1, j for some j. We have

It follows that $$\big\vert Q_{k,l} \cap U_{k+1}\big\vert \leq \alpha \vert Q_{k,l}\vert $$; thus, applying Lemma 7.2.1, we obtain

from which, summing over all l, we obtain

The latter gives w(U k ) ≤ β k w(U ₀). We also have | U k+1 | ≤ α | U k  | ; hence | U k  | → 0 as k → ∞. Therefore, the intersection of the U k ’s is a set of Lebesgue measure zero. We can therefore write

modulo a set of Lebesgue measure zero. Let us now find a γ > 0 such that the reverse Hölder inequality (7.2.4) holds. We have w(x) ≤ α k for almost all x in Q ∖ U k and therefore

provided γ > 0 is chosen small enough that $$(2^{n}\alpha ^{-1})^{\gamma }\beta < 1$$. Keeping track of the constants, we conclude the proof of the theorem with

(7.2.6)

and

which yields

(7.2.7)

Note that up to this point, α was an arbitrary number in (0, 1). □ 

Remark 7.2.3.

It is worth observing that for α such that $$(1-\alpha )^{p} = \frac{3} {4}$$, the constant γ in (7.2.6) decreases as $$[w]_{A_{p}}$$ increases, while the constant C in (7.2.7) increases as $$[w]_{A_{p}}$$ increases. This is because $$1 -\frac{3} {4}[w]_{A_{p}}^{-1} \geq \frac{1} {4}$$ and for $$t \in (\frac{1} {4},1)$$ the function $$\sqrt{t} - t$$ is decreasing. This allows us to obtain the following stronger version of Theorem 7.2.2: For any 1 ≤ p < ∞ and B > 1, there exist positive constants C = C(n, p, B) and γ = γ(n, p, B) such that for all w ∈ A p satisfying $$[w]_{A_{p}} \leq B$$ the reverse Hölder condition (7.2.4) holds for every cube Q. See Exercise 7.2.4(a) for details.

Observe that in the proof of Theorem 7.2.2 it was crucial to know that for some 0 < α, β < 1 we have

(7.2.8)

whenever S is a subset of the cube Q. No special property of Lebesgue measure was used in the proof of Theorem 7.2.2 other than its doubling property. Therefore, it is reasonable to ask whether Lebesgue measure in (7.2.8) can be replaced by a general measure μ satisfying the doubling property

(7.2.9)

for all cubes Q in R n . A straightforward adjustment of the proof of the previous theorem indicates that this is indeed the case.

Corollary 7.2.4.

Let w be a weight and let μ be a measure on R n satisfying (7.2.9). Suppose that there exist 0 < α,β < 1, such that

whenever S is a μ-measurable subset of a cube Q. Then there exist 0 < C,γ < ∞ [which depend only on the dimension n, the constant C n in (7.2.9), α, and β] such that for every cube Q in R n we have

(7.2.10)

Proof.

The proof of the corollary can be obtained almost verbatim from that of Theorem 7.2.2 by replacing Lebesgue measure with the doubling measure d μ and the constant 2 n by C n .

Precisely, we define $$\alpha _{k} = (C_{n}\alpha ^{-1})^{k}\alpha _{0}$$, where α ₀ is the μ-average of w over Q; then properties (1), (2), (3) concerning the selected cubes $$\{Q_{k,j}\}_{j}$$ are replaced by

(1 μ ) $$\alpha _{k} < \dfrac{1} {\mu (Q_{k,j})}\int _{Q_{k,j}}w(t)\,d\mu (t) \leq C_{n}\,\alpha _{k}$$.

(2 μ ) On Q ∖ U k we have w ≤ α k μ-almost everywhere, where $$U_{k} =\bigcup \limits _{j}Q_{k,j}.$$

(3 μ ) Each Q k+1, j is contained in some Q k, l .

To prove the upper inequality in (1 μ ) we use that the dyadic parent of each selected cube Q k, j was not selected and is contained in 3Q k, j . To prove (2 μ ) we need a differentiation theorem for doubling measures, analogous to that in Corollary 2.​1.​16.​ This can be found in Exercise 2.1.1. The remaining details of the proof are trivially adapted to the new setting. The conclusion is that for

(7.2.11)

and

(7.2.12)

(7.2.10) is satisfied. Notice that the choice of the constants (7.2.6) and (7.2.7) is valid in this case with C n in place of 2 n . □ 

7.2.2 Consequences of the Reverse Hölder Property

Having established the crucial reverse Hölder inequality for A p weights, we now pass to some very important applications. Among them, the first result of this section yields that an A p weight that lies a priori in L loc ¹(R n ) must actually lie in the better space L loc 1+σ (R n ) for some σ > 0 depending on the weight.

Theorem 7.2.5.

If w ∈ A p for some 1 ≤ p < ∞, then there exists a number γ > 0 (that depends on n, p, and $$[w]_{A_{p}}$$ ) such that $$w^{1+\gamma } \in A_{p}$$ .

Proof.

Let C be the constant in the proof of Theorem 7.2.2. When p = 1, we apply the reverse Hölder inequality of Theorem 7.2.2 to the weight w to obtain

for almost all x in the cube Q. Therefore, w 1+γ is an A ₁ weight with characteristic constant at most $$C^{1+\gamma }[w]_{A_{1}}^{1+\gamma }$$. When p > 1, there exist γ ₁, γ ₂ > 0 and C ₁, C ₂ > 0 such that the reverse Hölder inequality of Theorem 7.2.2 holds for the weights w ∈ A p and $$w^{- \frac{1} {p-1} } \in A_{p'}$$, that is,

Taking γ = min(γ ₁, γ ₂), both inequalities are satisfied with γ in the place of γ ₁, γ ₂. It follows that w 1+γ is in A p and satisfies

(7.2.13)

This concludes the proof of the theorem. □ 

Corollary 7.2.6.

For any 1 < p < ∞ and for every w ∈ A p there is a $$q = q(n,p,[w]_{A_{p}})$$ with q < p such that w ∈ A q . In other words, we have

Proof.

Given w ∈ A p , let γ, C ₁, C ₂ be as in the proof of Theorem 7.2.5. In view of the result in Exercise 7.1.3 with $$\delta = 1/(1+\gamma )$$, if w 1+γ  ∈ A p and

then w ∈ A q and

where the last estimate comes from (7.2.13). Since $$1 < q = \tfrac{p+\gamma } {1+\gamma } < p$$, the required conclusion follows. Observe that the constants C ₁ C ₂ p−1, q, and $$\frac{1} {\gamma }$$ increase as $$[w]_{A_{p}}$$ increases. □ 

Another powerful consequence of the reverse Hölder property of A p weights is the following characterization of all A ₁ weights.

Theorem 7.2.7.

Let w be an A ₁ weight. Then there exist $$0 <\varepsilon < 1$$ , a nonnegative function k such that k,k ⁻¹ ∈ L ∞ , and a nonnegative locally integrable function f that satisfies M(f) < ∞ a.e. such that

(7.2.14)

Conversely, given a nonnegative function k such that k,k ⁻¹ ∈ L ∞ and given a nonnegative locally integrable function f that satisfies M(f) < ∞ a.e., define w via (7.2.14) . Then w is an A ₁ weight that satisfies

(7.2.15)

where C n is a universal dimensional constant.

Proof.

In view of Theorem 7.2.2, there exist 0 < γ, C < ∞ such that the reverse Hölder condition

(7.2.16)

holds for all cubes Q and for all x in Q ∖ E Q , where E Q is a null subset of Q. We set

Letting N be the union of E Q over all Q with rational radii and centers in Q n , it follows from (7.2.16) that the uncentered Hardy–Littlewood maximal function M c (f) with respect to cubes satisfies

This implies that $$M(f) \leq C_{n}C^{1+\gamma }[w]_{A_{1}}^{1+\gamma }f$$ a.e. for some constant C n that depends only on the dimension. We now set

and we observe that $$C^{-1}C_{n}^{-\varepsilon }[w]_{A_{1}}^{-1} \leq k \leq 1$$ a.e.

It remains to prove the converse. Given a weight $$w = kM(f)^{\varepsilon }$$ in the form (7.2.14) and a cube Q, it suffices to show that

(7.2.17)

since then (7.2.15) follows trivially from (7.2.17) with $$w = kM(f)^{\varepsilon }$$ using that k, k ⁻¹ ∈ L ∞ . To prove (7.2.17), we write

Then

(7.2.18)

in view of Kolmogorov’s inequality (Exercise 2.1.5). But the last expression in (7.2.18) is at most a dimensional multiple of $$M(f)(x)^{\varepsilon }$$ for almost all x ∈ Q, which proves (7.2.17) when f is replaced by f χ 3Q on the left-hand side of the inequality. And for $$f\chi _{(3Q)^{c}}$$ we only need to notice that

for all x, t in Q, since any ball B centered at t that gives a nonzero average for $$f\chi _{(3Q)^{c}}$$ must have radius at least the side length of Q, and thus $$\sqrt{n}\,B$$ must also contain x. (Here $$\mathcal{M}$$ is the centered Hardy–Littlewood maximal operator introduced in Definition 2.​1.​1.​) Hence (7.2.17) also holds when f is replaced by $$f\chi _{(3Q)^{c}}$$ on the left-hand side. Combining these two estimates and using the subadditivity property $$M(f_{1} + f_{2})^{\varepsilon } \leq M(f_{1})^{\varepsilon } + M(f_{2})^{\varepsilon }$$, we obtain (7.2.17). □ 

We end this section with the following consequence of the reverse Hölder property of A p weights which can be viewed as a reverse property to (7.2.1).

Proposition 7.2.8.

Let 1 ≤ p < ∞ and w ∈ A p . Then there exist δ ∈ (0,1) and C > 0 depending only on n, p, and $$[w]_{A_{p}}$$ such that for any cube Q and any measurable subset S of Q we have

Proof.

Let C and γ be as in Theorem 7.2.2. We use Hölder’s inequality to write

where $$\delta = \frac{\gamma } {1+\gamma }$$. This proves the assertion. □ 

Exercises

7.2.1. Let w ∈ A p for some 1 < p < ∞ and let 1 ≤ q < ∞. Prove that the sublinear operator

is bounded on $$L^{p'q}(w)$$.

7.2.2. Let v be a real-valued locally integrable function on R n and let 1 < p < ∞. For a cube Q, let ν Q be the average of ν over Q.

(a) If e v is an A p weight, show that

(b) Conversely, if the preceding inequalities hold with some constant C in place of $$[\nu ]_{A_{p}}$$, then ν lies in A p with $$[\nu ]_{A_{p}} \leq C$$.

[Hint: Part (a): If e v  ∈ A p , use that

and obtain a similar estimate for the second quantity.]

7.2.3. This exercise assumes familiarity with the space B M O.

(a) Show that if $$\varphi \in A_{2}$$, then $$\log \varphi \in BMO$$ and $$\|\log \varphi \|_{BMO} \leq [\varphi ]_{A_{2}}$$.

(b) Prove that every B M O function is equal to a constant multiple of the logarithm of an A ₂ weight. Precisely, given f ∈ B M O show that

where $$c = 1/(2^{n+1}\|f\|_{BMO})$$.

(c) Prove that if $$\varphi$$ is in A p for some 1 < p < ∞, then $$\log \varphi$$ is in B M O by showing that

[Hint: Part (a): Use Exercise 7.2.2 with p = 2. Part (b): Use Exercise 7.2.2 and Corollary 3.1.7 in [131]. Use Part (c): Use that $$\varphi ^{- \frac{1} {p-1} } \in A_{p'}$$ when p > 2.]

7.2.4. Prove the following quantitative versions of Theorem 7.2.2 and Corollary 7.2.6.

(a) For any 1 ≤ p < ∞ and B > 1, there exists a positive constant C ₃(n, p, B) and γ = γ(n, p, B) such that for all w ∈ A p satisfying $$[w]_{A_{p}} \leq B$$, (7.2.4) holds for every cube Q with C ₃(n, p, B) in place of C.

(b) Given any 1 < p < ∞ and B > 1 there exists a constant C ₄(n, p, B) and δ = δ(n, p, B) such that for all w ∈ A p we have

7.2.5. Given a positive doubling measure μ on R n , define the characteristic constant $$[w]_{A_{p}(\mu )}$$ and the class A p (μ) for 1 < p < ∞.

(a) Show that statement (8) of Proposition 7.1.5 remains valid if Lebesgue measure is replaced by μ.

(b) Obtain as a consequence that if w ∈ A p (μ), then for all cubes Q and all μ-measurable subsets A of Q we have

Conclude that if Lebesgue measure is replaced by μ in Lemma 7.2.1, then the lemma is valid for w ∈ A p (μ).

(c) Use Corollary 7.2.4 to obtain that weights in A p (μ) satisfy a reverse Hölder condition.

(d) Prove that given a weight w ∈ A p (μ), there exists 1 < q < p, which depends on $$[w]_{A_{p}(\mu )}$$, such that w ∈ A q (μ).

7.2.6. Let 1 < q < ∞ and μ a positive measure on R n . We say that a positive function K on R n satisfies a reverse Hölder condition of order q with respect to μ, symbolically K ∈ R H q (μ) , if

For positive functions u, v on R n and 1 < p < ∞, show that

that is, v u ⁻¹ satisfies a reverse Hölder condition of order p′ with respect to u d x if and only if u v ⁻¹ is in A p (v d x). Conclude that

7.2.7. ([125]) Suppose that a positive function K on R n lies in R H p (d x) for some 1 < p < ∞. Show that there exists a δ > 0 such that K lies in R H p+δ (d x).

[Hint: By Exercise 7.2.6, K ∈ R H p (d x) is equivalent to the fact that K ⁻¹ ∈ A p′(K d x), and the index p′ can be improved by Exercise 7.2.5 (d).]

7.2.8. (a) Show that for any w ∈ A ₁ and any cube Q in R n and a > 1 we have

(b) Prove that there is a constant C n such that for all locally integrable functions f on R n and all cubes Q in R n we have

and an analogous statement is valid for M c .

[Hint: Part (a): Use (7.1.18). Part (b): Apply part (a) to $$M(f)^{\frac{1} {2} }$$, which is an A ₁ weight in view of Theorem 7.2.7.]

7.2.9. ([223]) For a weight w ∈ A ₁(R n ) define a quantity $$r = 1 + \frac{1} {2^{n+1}[w]_{A_{1}}}$$. Show that

[Hint: Fix a cube Q and consider the family $$\mathcal{F}_{Q}$$ of all cubes obtained by subdividing Q into a mesh of (2 n ) m subcubes of side length 2−m ℓ(Q) for all $$m = 1,2,\ldots$$. Define $$M_{Q}^{d}(f)(x) =\sup _{R\in \mathcal{F}_{Q},R\ni x}\vert R\vert ^{-1}\int _{R}\vert f\vert \,dy$$. Using Corollary 2.​1.​21obtain

for $$\lambda > w_{Q} = \frac{1} {\vert Q\vert }\int _{Q}w\,dt$$. Multiply by λ δ−1 and integrate to obtain

Replace w by w k  = min(k, w) and select $$\delta = \frac{1} {2^{n+1}[w]_{A_{1}}}$$ to deduce

using $$[w_{k}]_{A_{1}} \leq [w]_{A_{1}}$$. Then let k → ∞.]

7.2.10. Let 1 < p < ∞. Recall that a pair of weights (u, w) that satisfies

is said to be of class (A p , A p ) . The quantity $$[u,w]_{(A_{p},A_{p})}$$ is called the (A p , A p ) characteristic constant of the pair (u, w).

(a) Show that for any g ∈ L loc ¹(R n ) with 0 < g < ∞ a.e., the pair (g, M(g)) is of class (A p , A p ) with characteristic constant independent of f.

(b) If (u, w) is of class (A p , A p ), then the Hardy–Littlewood maximal operator M may not map L p (w) to L p (u).

(c) Given g ∈ L loc ¹(R n ) with 0 < g < ∞ a.e., conclude that Hardy–Littlewood maximal operator M maps L p (M(g)d x) to L p, ∞ (g d x) and also L q (M(g)d x) to L q (g d x) for any q with p < q < ∞.

[Hint: Part (a): Use Hölder’s inequality and Theorem 7.2.7. Part (b): Try the pair $$\big(M(g)^{1-p},\vert g\vert ^{1-p}\big)$$ for a suitable g. Part (c): Use Exercises 7.1.6 and 7.1.7.]

7.3 The A ∞ Condition

In this section we examine more closely the class of all A p weights. It turns out that A p weights possess properties that are p-independent but delicate enough to characterize them without reference to a specific value of p. The A p classes increase as p increases, and it is only natural to consider their limit as p → ∞. Not surprisingly, a condition obtained as a limit of the A p conditions as p → ∞ provides some unexpected but insightful characterizations of the class of all A p weights.

7.3.1 The Class of A ∞ Weights

Let us start by recalling a simple consequence of Jensen’s inequality:

(7.3.1)

which holds for all measurable functions h on a probability space (X, μ) and all 0 < q < ∞. See Exercise 1.1.3(b). Moreover, part (c) of the same exercise says that the limit of the expressions on the left in (7.3.1) as q → 0 is equal to the expression on the right in (7.3.1).

We apply (7.3.1) to the function $$h = w^{-1}$$ for some weight w in A p with $$q = 1/(p - 1)$$. We obtain

(7.3.2)

and the limit of the expressions on the left in (7.3.2) as p → ∞ is equal to the expression on the right in (7.3.2). This observation provides the motivation for the following definition.

Definition 7.3.1.

A weight w is called an A ∞ weight if

The quantity $$[w]_{A_{\infty }}$$ is called the A ∞ characteristic constant of w.

It follows from the previous definition and (7.3.2) that for all 1 ≤ p < ∞ we have

This means that

(7.3.3)

but the remarkable thing is that equality actually holds in (7.3.3), a deep property that requires some work.

Before we examine this and other characterizations of A ∞ weights, we discuss some of their elementary properties.

Proposition 7.3.2.

Let w ∈ A ∞ . Then

(1)

$$[\delta ^{\lambda }(w)]_{A_{\infty }} = [w]_{A_{\infty }}$$ , where $$\delta ^{\lambda }(w)(x) = w(\lambda x_{1},\ldots,\lambda x_{n})$$ and λ > 0.

(2)

$$[\tau ^{z}(w)]_{A_{\infty }} = [w]_{A_{\infty }}$$ , where $$\tau ^{z}(w)(x) = w(x - z)$$ , z ∈ R n .

(3)

$$[\lambda w]_{A_{\infty }} = [w]_{A_{\infty }}$$ for all λ > 0.

(4)

$$[w]_{A_{\infty }} \geq 1$$ .

(5)

The following is an equivalent characterization of the A ∞ characteristic constant of w:

(6)

The measure w(x) dx is doubling; precisely, for all λ > 1 and all cubes Q we have

As usual, λQ here denotes the cube with the same center as Q and side length λ times that of Q.

We note that estimate (6) is not as good as λ → ∞ but it can be substantially improved using the case λ = 2. We refer to Exercise 7.3.1 for an improvement.

Proof.

Properties (1)–(3) are elementary, while property (4) is a consequence of Exercise 1.1.3(b). To show (5), first observe that by taking $$f = w^{-1}$$, the expression on the right in (5) is at least as big as $$[w]_{A_{\infty }}$$. Conversely, (7.3.1) gives

which, after a simple algebraic manipulation, can be written as

whenever f does not vanish almost everywhere on Q. Taking the supremum over all such f and all cubes Q in R n , we obtain that the expression on the right in (5) is at most $$[w]_{A_{\infty }}$$.

To prove the doubling property for A ∞ weights, we fix λ > 1 and we apply property (5) to the cube λ Q in place of Q and to the function

(7.3.4)

where c is chosen so that $$c^{1/\lambda ^{n} } = 2[w]_{A_{\infty }}$$. We obtain

which implies (6) if we take into account the chosen value of c. □ 

7.3.2 Characterizations of A ∞ Weights

Having established some elementary properties of A ∞ weights, we now turn to some of their deeper properties, one of which is that every A ∞ weight lies in some A p for p < ∞. It also turns out that A ∞ weights are characterized by the reverse Hölder property, which as we saw is a fundamental property of A p weights. The following is the main theorem of this section.

Theorem 7.3.3.

Suppose that w is a weight. Then w is in A ∞ if and only if any one of the following conditions holds:

(a) There exist 0 < γ,δ < 1 such that for all cubes Q in R n we have

(b) There exist 0 < α,β < 1 such that for all cubes Q and all measurable subsets A of Q we have

(c) The reverse Hölder condition holds for w, that is, there exist $$0 < C_{1},\varepsilon < \infty $$ such that for all cubes Q we have

(d) There exist $$0 < C_{2},\varepsilon _{0} < \infty $$ such that for all cubes Q and all measurable subsets A of Q we have

(e) There exist 0 < α′,β′ < 1 such that for all cubes Q and all measurable subsets A of Q we have

(f) There exist p,C ₃ < ∞ such that $$[w]_{A_{p}} \leq C_{3}$$ . In other words, w lies in A p for some p ∈ [1,∞).

All the constants $$C_{1},C_{2},C_{3},\alpha,\beta,\gamma,\delta,\alpha ',\beta ',\varepsilon,\varepsilon _{0}$$ , and p in (a)–(f) depend only on the dimension n and on $$[w]_{A_{\infty }}$$ . Moreover, if any of the statements in (a)–(f) is valid, then so is any other statement in (a)–(f) with constants that depend only on the dimension n and the constants that appear in the assumed statement.

Proof.

The proof follows from the sequence of implications

At each step we keep track of the way the constants depend on the constants of the previous step. This is needed to validate the last assertion of the theorem.

Fix a cube Q. Since multiplication of an A ∞ weight with a positive scalar does not alter its A ∞ characteristic, we may assume that $$\int _{Q}\log w(t)\,dt = 0$$. This implies that $$\mathop{\mbox{ Avg}}_{Q}w \leq [w]_{A_{\infty }}$$. Then we have

which proves (a ) with $$\gamma = [w]_{A_{\infty }}^{-1}(e^{2[w]_{A_{\infty }}}- 1)^{-1}$$ and $$\delta = \frac{1} {2}$$.

Let Q be fixed and let A be a subset of Q with w(A) > β w(Q) for some β to be chosen later. Setting S = Q ∖ A, we have w(S) < (1 −β)w(Q). We write $$S = S_{1} \cup S_{2}$$, where

For S ₂ we have | S ₂ | ≤ δ | Q | by assumption (a ). For S ₁ we use Chebyshev’s inequality to obtain

Adding the estimates for | S ₁ | and | S ₂ | , we obtain

Choosing numbers α, β in (0, 1) such that $$\delta +\frac{1-\beta } {\gamma } = 1-\alpha$$, for example $$\alpha = \frac{1-\delta } {2}$$ and $$\beta = 1 -\frac{(1-\delta )\gamma } {2}$$, we obtain | S | ≤ (1 −α) | Q | , that is, | A |  > α | Q | .

This was proved in Corollary 7.2.4. To keep track of the constants, we note that the choices

as given in (7.2.6) and (7.2.7) serve our purposes.

We apply first Hölder’s inequality with exponents $$1+\varepsilon$$ and $$(1+\varepsilon )/\varepsilon$$ and then the reverse Hölder estimate to obtain

which gives

This proves (d ) with $$\varepsilon _{0} = \frac{\varepsilon } {1+\varepsilon }$$ and C ₂ = C ₁.

Pick an 0 < α″ < 1 small enough that $$\beta '' = C_{2}(\alpha '')^{\varepsilon _{0}} < 1$$. It follows from (d ) that

(7.3.5)

for all cubes Q and all A measurable subsets of Q. Replacing A by Q ∖ A, the implication in (7.3.5) can be equivalently written as

In other words, for measurable subsets A of Q we have

(7.3.6)

which is the statement in (e ) if we set $$\alpha ' = (1 -\beta '')$$ and $$\beta ' = 1 -\alpha ''$$. Note that (7.3.5) and (7.3.6) are indeed equivalent.

We begin by examining condition (e ), which can be written as

or, equivalently, as

after defining the measure d μ(t) = w(t) d t. As we have already seen, the assertions in (7.3.5) and (7.3.6) are equivalent. Therefore, we may use Exercise 7.3.2 to deduce that the measure μ is doubling, i.e., it satisfies property (7.2.9) for some constant C n  = C n (α′, β′), and hence the hypotheses of Corollary 7.2.4 are satisfied. We conclude that the weight w ⁻¹ satisfies a reverse Hölder estimate with respect to the measure μ, that is, if γ, C are defined as in (7.2.11) and (7.2.12) [in which α is replaced by α′, β by β′, and C n is the doubling constant of w(x) d x], then we have

(7.3.7)

for all cubes Q in R n . Setting $$p = 1 + \frac{1} {\gamma }$$ and raising to the pth power, we can rewrite (7.3.7) as the A p condition for w. We can therefore take C ₃ = C p to conclude the proof of (f).

This is trivial, since $$[w]_{A_{\infty }} \leq [w]_{A_{p}}$$. □ 

An immediate consequence of the preceding theorem is the following result relating A ∞ to A p .

Corollary 7.3.4.

The following equality is valid:

Exercises

7.3.1. Let λ > 0, Q be a cube in R n , and w ∈ A ∞ (R n ).

(a) Show that property (6) in Proposition 7.3.2 can be improved to

(b) Prove that

[Hint: Part (a): Take c in (7.3.4) such that $$c^{1/\lambda ^{n} } = (1+\varepsilon )[w]_{A_{\infty }}$$. Part (b): Use the estimate in property (6) of Proposition 7.3.2 with λ = 2.]

7.3.2. Suppose that μ is a positive Borel measure on R n with the property that for all cubes Q and all measurable subsets A of Q we have

for some fixed 0 < α, β < 1. Show that μ is doubling [i.e., it satisfies (7.2.9)].

[Hint: Use that $$\vert S\vert > (1-\alpha )\vert Q\vert \Rightarrow \mu (S) > (1-\beta )\mu (Q)$$ when $$S \subseteq Q$$.]

7.3.3. Prove that a weight w is in A p if and only if both w and $$w^{- \frac{1} {p-1} }$$ are in A ∞ .

[Hint: You may want to use the result of Exercise 7.2.2.]

7.3.4. ([33], [343]) Prove that if P(x) is a polynomial of degree k in R n , then log | P(x) | is in B M O with norm depending only on k and n and not on the coefficients of the polynomial.

[Hint: Use that all norms on the finite-dimensional space of polynomials of degree at most k are equivalent to show that | P(x) | satisfies a reverse Hölder inequality. Therefore, | P(x) | is an A ∞ weight and thus Exercise 7.2.3 (c) is applicable.]

7.3.5. Show that the product of two A ₁ weights may not be an A ∞ weight.

7.3.6. Let g be in L p (w) for some 1 ≤ p ≤ ∞ and w ∈ A p . Prove that g ∈ L loc ¹(R n ).

[Hint: Let B be a ball. In the case p < ∞, write $$\int _{B}\vert g\vert \,dx =\int _{B}(\vert g\vert w^{-\frac{1} {p} })w^{\frac{1} {p} }\,dx$$ and apply Hölder’s inequality. In the case p = ∞, use that $$w \in A_{p_{0}}$$ for some p ₀ < ∞.]

7.3.7. ([278]) Show that a weight w lies in A ∞ if and only if there exist γ, C > 0 such that for all cubes Q we have

for all $$\lambda >\mathop{ \mbox{ Avg}}_{Q}w$$.

[Hint: The displayed condition easily implies that

where k > 0, w k  = min(w, k) and $$\delta =\varepsilon /(1+\varepsilon )$$. Take $$\varepsilon > 0$$ small enough to obtain the reverse Hölder condition (c ) in Theorem 7.3.3 for w k . Let k → ∞ to obtain the same conclusion for w. Conversely, find constants γ, δ ∈ (0, 1) as in condition (a) of Theorem 7.3.3 and for $$\lambda >\mathop{ \mbox{ Avg}}_{Q}w$$ write the set $$\{w >\lambda \} \cap Q$$ as a union of maximal dyadic cubes Q j such that $$\lambda <\mathop{ \mbox{ Avg}}_{Q_{j}}w \leq 2^{n}\lambda$$ for all j. Then $$w(Q_{j}) \leq 2^{n}\lambda \vert Q_{j}\vert \leq \frac{2^{n}\lambda } {1-\delta }\vert Q_{j} \cap \{ w >\gamma \lambda \} \vert $$ and the required conclusion follows by summing on j.]

7.4 Weighted Norm Inequalities for Singular Integrals

We now address a topic of great interest in the theory of singular integrals, their boundedness properties on weighted L p spaces. It turns out that a certain amount of regularity must be imposed on the kernels of these operators to obtain the aforementioned weighted estimates.

7.4.1 Singular Integrals of Non Convolution type

We introduce some definitions.

Definition 7.4.1.

Let $$0 <\delta,A < \infty $$. A function K(x, y) defined for x, y ∈ R n with x ≠ y is called a standard kernel (with constants δ and A) if

(7.4.1)

and whenever $$\vert x - x'\vert \leq \frac{1} {2}\max \big(\vert x - y\vert,\vert x' - y\vert \big)$$ we have

(7.4.2)

and also when $$\vert y - y'\vert \leq \frac{1} {2}\max \big(\vert x - y\vert,\vert x - y'\vert \big)$$ we have

(7.4.3)

The class of all kernels that satisfy (7.4.1), (7.4.2), and (7.4.3) is denoted by S K(δ, A).

Definition 7.4.2.

Let 0 < δ, A < ∞ and K in S K(δ, A). A Calderón–Zygmund operator associated with K is a linear operator T defined on $$\mathcal{S}(\mathbf{R}^{n})$$ that admits a bounded extension on L ²(R n ),

(7.4.4)

and that satisfies

(7.4.5)

for all $$f \in \mathcal{C}_{0}^{\infty }$$ and x not in the support of f. The class of all Calderón–Zygmund operators associated with kernels in S K(δ, A) that are bounded on L ² with norm at most B is denoted by C Z O(δ, A, B). Note that there is no unique T associated with a given K. Given a Calderón–Zygmund operator T in C Z O(δ, A, B), we define the truncated operator $$T^{(\varepsilon )}$$ as

and the maximal operator associated with T as follows:

We note that if T is in C Z O(δ, A, B), then $$T^{(\varepsilon )}(f)$$ and T (∗)(f) are well defined for all f in $$\bigcup _{1\leq p<\infty }L^{p}(\mathbf{R}^{n})$$. It is also well defined whenever f is locally integrable and satisfies $$\int _{\vert x-y\vert \geq \varepsilon }\vert f(y)\vert \,\vert x - y\vert ^{-n}dy < \infty $$ for all x ∈ R n and $$\varepsilon > 0$$.

The class of kernels in S K(δ, A) extends the family of convolution kernels that satisfy conditions (5.​3.​10), (5.​3.​11), and (5.​3.​12). Obviously, the associated operators in C Z O(δ, A, B) generalize the associated convolution operators.

A fundamental property of operators in C Z O(δ, A, B) is that they have bounded extensions on all the L p (R n ) spaces and also from L ¹(R n ) to weak L ¹(R n ). This is proved via an adaptation of Theorem 5.​3.​3; see Theorem 4.2.2 in [131]. There are analogous results for the maximal counterparts T (∗) of elements of C Z O(δ, A, B). In fact, an analogue of Theorem 5.​3.​5 yields that T (∗) is L p bounded for 1 < p < ∞ and weak type (1, 1); this result is contained in Theorem 4.2.4 in [131].

We discuss weighted inequalities for singular integrals for general operators in C Z O(δ, A, B). In Subsections 7.4.2 and 7.4.3, the reader may wish to replace kernels in S K(δ, A) by the more familiar functions K(x) defined on R n ∖{0} that satisfy (5.​3.​10), (5.​3.​11), and (5.​3.​12).

7.4.2 A Good Lambda Estimate for Singular Integrals

The following theorem is the main result of this section.

Theorem 7.4.3.

Let 1 ≤ p ≤∞, w ∈ A p , and T in CZO(δ,A,B). Then there exist positive constants ¹ $$C_{0} = C_{0}(n,p,[w]_{A_{p}})$$ , $$\varepsilon _{0} =\varepsilon _{0}(n,p,[w]_{A_{p}})$$ , and c ₀ (n,δ), such that if $$\gamma _{0} = c_{0}(n,\delta )/A$$ , then for all 0 < γ < γ ₀ we have

(7.4.6)

for all locally integrable functions f for which

for all x ∈ R n and $$\varepsilon > 0$$ . Here M denotes the Hardy–Littlewood maximal operator.

Proof.

We write the open set

where Q j are the Whitney cubes (see Appendix J). We set

where a Q denotes the cube with the same center as Q whose side length is a ℓ(Q), where ℓ(Q) is the side length of Q. We note that in view of the properties of the Whitney cubes, the distance from Q j to $$\varOmega ^{c}$$ is at most $$4\sqrt{n}\,\ell(Q_{j})$$. But the distance from Q j to the boundary of Q j ∗ is $$(5\sqrt{n} -\frac{1} {2})\,\ell(Q_{j})$$, which is bigger than $$4\sqrt{n}\,\ell(Q_{j})$$. Therefore, Q j ∗ must meet $$\varOmega ^{c}$$ and for every cube Q j we fix a point y j in $$\varOmega ^{c} \cap Q_{j}^{{\ast}}$$. See Figure 7.1.

Fig. 7.1

A picture of the proof.

We also fix f in $$\bigcup _{1\leq p<\infty }L^{p}(\mathbf{R}^{n})$$, and for each j we write $$f = f_{0}^{j} + f_{\infty }^{j},$$ where $$f_{0}^{j} = f\chi _{Q_{j}^{{\ast}{\ast}}}$$ is the part of f near Q j and $$f_{\infty }^{j} = f\chi _{(Q_{j}^{{\ast}{\ast}})^{c}}$$ is the part of f away from Q j .

We now claim that the following estimate is true:

(7.4.7)

Once the validity of (7.4.7) is established, we apply Theorem 7.3.3 (d) when p = ∞ or Proposition 7.2.8 when p < ∞ to obtain constants $$\varepsilon _{0},C_{2} > 0$$, which depend on $$[w]_{A_{p}}$$, p, n when p < ∞ and on $$[w]_{A_{\infty }}$$ and n when p = ∞, such that

Then a simple summation in j gives (7.4.6) with $$C_{0} = C_{2}(C_{n})^{\varepsilon _{0}}$$, and recall that C ₂ and $$\varepsilon _{0}$$ depend on n and $$[w]_{A_{p}}$$ and on p if p < ∞.

In proving estimate (7.4.7), we may assume that for each cube Q j there exists a z j  ∈ Q j such that M(f)(z j ) ≤ γ λ; otherwise, the set on the left in (7.4.7) is empty.

We now invoke Theorem 4.2.4 in [131], which states that T (∗) maps L ¹(R n ) to L 1, ∞ (R n ) with norm at most C(n)(A + B). We have the estimate

(7.4.8)

where

To control I ₀ λ we note that f ₀ j is in L ¹(R n ) and we argue as follows:

(7.4.9)

Next we claim that I ∞ λ  = 0 if we take γ sufficiently small. We first show that for all x ∈ Q j we have

(7.4.10)

Indeed, let us fix an $$\varepsilon > 0$$. We have

where

in view of identity (5.​4.​7).

We now make a couple of observations. For t∉Q j ∗∗, x, z j  ∈ Q j , and y j  ∈ Q j ∗ we have

(7.4.11)

Indeed,

and

Using this estimate and the inequalities

we obtain the first estimate in (7.4.11). Likewise, we have

and

and these give

yielding the second estimate in (7.4.11).

Since $$\vert x - y_{j}\vert \leq \frac{1} {2}\vert t - y_{j}\vert \leq \frac{1} {2}\max \big(\vert t - x\vert,\vert t - y_{j}\vert \big)$$, we have

hence, we obtain

using Theorem 2.​1.​10.​ Using (7.4.11) we deduce

Again using (7.4.11), we obtain

This proves (7.4.10) with constant $$C_{n,\delta }^{(1)} = C_{n,\delta }'' + C_{n}' + C_{n}''$$.

Having established (7.4.10), we next claim that

(7.4.12)

To prove (7.4.12) we fix a cube Q j and $$\varepsilon > 0$$. We let R j be the smallest number such that

See Figure 7.2. We consider the following two cases.

Fig. 7.2

The ball B(y j , R j ).

Case (1): $$\,\,\,\,\varepsilon \geq R_{j}$$. Since $$Q_{j}^{{\ast}{\ast}}\subseteq B(y_{j},\varepsilon )$$, we have $$B(y_{j},\varepsilon )^{c} \subseteq (Q_{j}^{{\ast}{\ast}})^{c}$$ and therefore

so (7.4.12) holds easily in this case.

Case (2): $$\,\,\,\,0 <\varepsilon < R_{j}$$. Note that if t ∈ (Q j ∗∗) c , then | t − y j  | ≥ 40 n ℓ(Q j ). On the other hand, $$R_{j} \leq \text{diam}(Q_{j}^{{\ast}{\ast}}) = 100\,n^{\frac{3} {2} }\,\ell(Q_{j})$$. This implies that

Notice also that in this case we have $$B(y_{j},R_{j})^{c} \subseteq (Q_{j}^{{\ast}{\ast}})^{c}$$, hence

Therefore, we have

where in the penultimate estimate we used (7.4.11). The proof of (7.4.12) follows with the required bound C n ⁽²⁾ A.

Combining (7.4.10) and (7.4.12), we obtain

Recalling that $$y_{j}\notin \varOmega$$ and that M(f)(z j ) ≤ γ λ, we deduce

Setting $$\gamma _{0} =\big (C_{n,\delta }^{(1)} + C_{n}^{(2)}\big)^{-1}\,A^{-1} = c_{0}(n,\delta )A^{-1}$$, for 0 < γ < γ ₀, we have that the set

is empty. This shows that the quantity I ∞ γ vanishes if γ is smaller than γ ₀. Returning to (7.4.8) and using the estimate (7.4.9) proved earlier, we conclude the proof of (7.4.7), which, as indicated earlier, implies the theorem. □ 

Remark 7.4.4.

We observe that for any δ > 0, estimate (7.4.6) also holds for the operator

(7.4.13)

with the same constant (which is independent of δ).

To see the validity of (7.4.6) for T δ (∗), it suffices to prove

(7.4.14)

which is a version of (7.4.12) with T (∗) replaced by T δ (∗). The following cases arise:

Case (1 ′ ): $$R_{j} \leq \delta \leq \varepsilon$$ or $$\delta \leq R_{j} \leq \varepsilon$$. Here, as in Case (1) we have

Case (2 ′ ): $$\delta \leq \varepsilon < R_{j}$$. As in Case (2) we have

thus

As in the proof of Case (2), we bound the first term on the right of the last displayed expression by C n ⁽²⁾ A M(f)(z j ) while the second term is at most T δ (∗)(f)(y j ).

7.4.3 Consequences of the Good Lambda Estimate

Having obtained the important good lambda weighted estimate for singular integrals, we now pass to some of its consequences. We begin with the following lemma:

Lemma 7.4.5.

Let 1 ≤ p < ∞, $$\varepsilon > 0$$ , w ∈ A p , x ∈ R n , and f ∈ L p (w). Then we have

for some constant C ₀₀ depending on the stated parameters. In particular $$T^{(\varepsilon )}(f)$$ and T (∗) (f) are defined for f ∈ L p (w).

Proof.

For each $$\varepsilon > 0$$ and x pick a cube $$Q_{0} = Q_{0}(x,\varepsilon )$$ of side length $$c_{n}\varepsilon$$ (for some constant c n ) such that $$Q_{0} \subseteq B(x,\varepsilon )$$. Set Q j  = 2 j Q ₀ for j ≥ 0. We have

But Proposition 7.2.8 gives for some $$\delta =\delta (n,p,[w]_{A_{p}})$$ that

from which it follows that

In view of this estimate, the previous series converges. Note that C′ and thus C ₀₀ depend on $$[w]_{A_{p}},n,p,x,\varepsilon$$, and w(Q ₀).

This argument is also valid in the case p = 1 by an obvious modification. □ 

Theorem 7.4.6.

Let A,B,β > 0 and let T be a CZO(β,A,B). Then given 1 < p < ∞, there is a constant $$C_{p} = C_{p}(n,\beta,[w]_{A_{p}})$$ such that

(7.4.15)

for all w ∈ A p and f ∈ L p (w). There is also a constant $$C_{1} = C_{1}(n,\beta,[w]_{A_{1}})$$ such that

(7.4.16)

for all w ∈ A ₁ and f ∈ L ¹ (w).

Proof.

This theorem is a consequence of the estimate proved in the previous theorem. For technical reasons, it is useful to fix a δ > 0 and work with the auxiliary maximal operator T δ (∗) defined in (7.4.13) instead of T (∗). We begin by taking 1 < p < ∞ and f ∈ L p (w) for some w ∈ A p . We write

which we control by

Using Theorem 7.4.3 (or rather Remark 7.4.4), there are $$C_{0} = C_{0}(n,[w]_{A_{p}})$$, $$\varepsilon _{0} =\varepsilon _{0}(n,[w]_{A_{p}})$$, and $$\gamma _{0} = c_{0}(n,\beta )A^{-1}$$, such that the preceding displayed expression is bounded by

which is equal to

Taking $$\gamma =\min \big (\frac{1} {2}c_{0}(n,\beta )A^{-1}, \frac{1} {2}(2C_{0}3^{p})^{-\frac{1} {\varepsilon _{0}} }(A\! +\! B)^{-1}\big) <\gamma _{0}$$, we conclude that

(7.4.17)

We now prove a similar estimate when p = 1. For f ∈ L ¹(w) and w ∈ A ₁ we have

and this expression is controlled by

Recalling that $$\gamma _{0} = c_{0}(n,\beta )A^{-1}$$ and choosing $$\gamma =\min \big (\frac{1} {2}\gamma _{0}, \frac{1} {2}(6C_{0})^{-\frac{1} {\varepsilon _{0}} }(A\! +\! B)^{-1}\big)$$, it follows that

(7.4.18)

Estimate (7.4.15) would follow from (7.4.17) if we knew that $$\|T_{\delta }^{({\ast})}(f)\|_{L^{p}(w)} < \infty $$ whenever 1 < p < ∞, w ∈ A p and f ∈ L p (w), while (7.4.16) would follow from (7.4.18) if we had $$\|T_{\delta }^{({\ast})}(f)\|_{L^{1,\infty }(w)} < \infty $$ whenever w ∈ A ₁ and f ∈ L ¹(w). Since we do not know that these quantities are finite, a certain amount of work is needed.

To deal with this problem we momentarily restrict attention to a special class of functions on R n , the class of bounded functions with compact support. Such functions are dense in L p (w) when w ∈ A p and 1 ≤ p < ∞; see Exercise 7.4.1. Let h be a bounded function with compact support on R n . Then $$T_{\delta }^{({\ast})}(h) \leq C_{1}\delta ^{-n}\|h\|_{L^{1}}$$ and $$T_{\delta }^{({\ast})}(h)(x) \leq C_{2}(h)\vert x\vert ^{-n}$$ for x away from the support of h. It follows that

for all x ∈ R n . Furthermore, if h is nonzero, then

and therefore for w ∈ A ₁,

while for 1 < p < ∞ and w ∈ A p ,

in view of Theorem 7.1.9. Using these facts, (7.4.17), (7.4.18), and Theorem 7.1.9 once more, we conclude that for all δ > 0 and 1 < p < ∞ we have

(7.4.19)

whenever h a bounded function with compact support. The constants $$\widetilde{C}_{p}$$, $$\widetilde{C}_{p}'$$, and C p depend only on the parameters n, β, p, and $$[w]_{A_{p}}$$.

We now extend estimates (7.4.16) and (7.4.15) to functions in L p (R n , w d x). Given 1 ≤ p < ∞, w ∈ A p , and f ∈ L p (w), let

Then f N is a bounded function with compact support that converges to f in L p (w) (i.e., $$\|f_{N} - f\|_{L^{p}(w)} \rightarrow 0$$ as N → ∞) by the Lebesgue dominated convergence theorem. Also | f N  | ≤ | f | for all N. Sublinearity and Lemma 7.4.5 give for all x ∈ R n ,

and this converges to zero as N → ∞ since C ₀₀(w, n, p, x, δ) < ∞. Therefore

pointwise, and Fatou’s lemma for weak type spaces [see Exercise 1.1.12 (d)] gives for w ∈ A ₁ and f ∈ L ¹(w),

since | f N  | ≤ | f | for all N. An analogous argument gives the estimate

for w ∈ A p and f ∈ L p (w) when 1 < p < ∞.

It remains to prove (7.4.15) and (7.4.16) for T (∗). But this is also an easy consequence of Fatou’s lemma, since the constants C p and C ₁ are independent of δ and

for all f ∈ L p (w). □ 

We end this subsection by making the comment that if a given T in C Z O(δ, A, B) is pointwise controlled by T (∗), then the estimates of Theorem 7.4.6 also hold for it. This is the case for the Hilbert transform, the Riesz transforms, and other classical singular integral operators.

7.4.4 Necessity of the A p Condition

We have established the main theorems relating Calderón–Zygmund operators and A p weights, namely that such operators are bounded on L p (w) whenever w lies in A p . It is natural to ask whether the A p condition is necessary for the boundedness of singular integrals on L p . We end this section by indicating the necessity of the A p condition for the boundedness of the Riesz transforms on weighted L p spaces.

Theorem 7.4.7.

Let w be a weight in R n and let 1 ≤ p < ∞. Suppose that each of the Riesz transforms R j is of weak type (p,p) with respect to w. Then w must be an A p weight. Similarly, let w be a weight in $$\mathbf{R}$$ . If the Hilbert transform H is of weak type (p,p) with respect to w, then w must be an A p weight.

Proof.

We prove the n-dimensional case, n ≥ 2. The one-dimensional case is essentially contained in following argument, suitably adjusted.

Let Q be a cube and let f be a nonnegative function on R n supported in Q that satisfies $$\mathop{\mbox{ Avg}}_{Q}f > 0$$. Let Q′ be the cube that shares a corner with Q, has the same length as Q, and satisfies x j  ≥ y j for all 1 ≤ j ≤ n whenever x ∈ Q′ and y ∈ Q. Then for x ∈ Q′ we have

But if x ∈ Q′ and y ∈ Q we must have that $$\vert x - y\vert \leq 2\sqrt{n}\,\ell(Q)$$, which implies that $$\vert x - y\vert ^{-n} \geq (2\sqrt{n})^{-n}\vert Q\vert ^{-1}$$. Let $$C_{n} =\varGamma (\frac{n+1} {2} )(2\sqrt{n})^{-n}\pi ^{-\frac{n+1} {2} }$$. It follows that for all $$0 <\alpha < C_{n}\mathop{ \mbox{ Avg}}_{Q}f$$ we have

Since the operator $$\sum _{j=1}^{n}R_{j}$$ is of weak type (p, p) with respect to w (with constant C), we must have

for all $$\alpha < C_{n}\mathop{ \mbox{ Avg}}_{Q}f$$, which implies that

(7.4.20)

We observe that we can reverse the roles of Q and Q′ and obtain

(7.4.21)

for all g supported in Q′. In particular, taking g = χ Q′ in (7.4.21) gives that

Using this estimate and (7.4.20), we obtain

(7.4.22)

Using the characterization of the A p characteristic constant in Proposition 7.1.5 (8), it follows that

hence w ∈ A p . □ 

Exercises

7.4.1. Let 1 ≤ p < ∞ and let w ∈ L loc ¹(R n ) satisfy w > 0 a.e. Show that $$\mathcal{C}_{0}^{\infty }(\mathbf{R}^{n})$$ is dense in L p (w). In particular this assertion holds for any w ∈ A ∞ .

7.4.2. ([74]) Let T be in C Z O(δ, A, B). Show that for all $$\varepsilon > 0$$ and all 1 < p < ∞ there exists a constant $$C_{n,p,\varepsilon,\delta }$$ such that for all $$f \in L^{p}(\mathbf{R}^{n})$$ and for all measurable nonnegative functions u with $$u^{1+\varepsilon } \in L_{\mbox{ loc}}^{1}(\mathbf{R}^{n})$$ and $$M(u^{1+\varepsilon }) < \infty $$ a.e. we have

[Hint: Obtain this result as a consequence of Theorems 7.4.6 and 7.2.7.]

7.4.3. Use the idea of the proof of Theorem 7.4.6 to prove the following result. Suppose that for some fixed A, B > 0 the nonnegative μ-measurable functions F and G on a σ-finite measure space (X, μ) satisfy the distributional inequality

for all α > 0. Given 0 < p < ∞, if A < B p and $$\|G\|_{L^{p}(\mu )} < \infty $$, show that

7.4.4. Let α > 0, w ∈ A ₁, and $$f \in L^{1}(\mathbf{R}^{n},w) \cap L^{1}(\mathbf{R}^{n})$$. Let $$f = g + b$$ be the Calderón–Zygmund decomposition of f at height α > 0 given in Theorem 5.​3.​1, such that $$b =\sum _{j}b_{j}$$, where each b j is supported in a dyadic cube Q j , $$\int _{Q_{j}}b_{j}(x)dx = 0,$$ and Q j and Q k have disjoint interiors when j ≠ k. Prove that

(a)

$$\|g\|_{L^{1}(w)} \leq [w]_{A_{1}}\|f\|_{L^{1}(w)}$$ and $$\|g\|_{L^{\infty }(w)} =\| g\|_{L^{\infty }} \leq 2^{n}\alpha$$,

(b)

$$\|b_{j}\|_{L^{1}(w)} \leq (1 + [w]_{A_{1}})\|f\|_{L^{1}(Q_{j},w)}$$ and $$\|b\|_{L^{1}(w)} \leq (1 + [w]_{A_{1}})\|f\|_{L^{1}(w)},$$

(c)

7.4.5. Assume that T is an operator associated with a kernel in S K(δ, A). Suppose that T maps L ²(w) to L ²(w) for all w ∈ A ₁ with bound B w . Prove that there is a constant C n, δ such that

for all w ∈ A ₁.

[Hint: Apply the idea of the proof of Theorem 5.​3.​3 using the Calderón-Zygmund decomposition $$f = g + b$$ of Exercise 7.4.4 at height γ α for a suitable γ. To estimate T(g) use an L ²(w) estimate and Exercise 7.4.4. To estimate T(b) use the mean value property, the fact that

and Exercise 7.4.4 to obtain the required estimate.]

7.4.6. Recall that the transpose T t of a linear operator T is defined by

for all suitable f and g. Suppose that T is a linear operator that maps L p (R n , v d x) to itself for some 1 < p < ∞ and some v ∈ A p . Show that the transpose operator T t maps L p′(R n , w d x) to itself with the same norm, where $$w = v^{1-p'} \in A_{p'}$$.

7.4.7. Suppose that T is a linear operator that maps $$L^{2}(\mathbf{R}^{n},vdx)$$ to itself for all v such that v ⁻¹ ∈ A ₁. Show that the transpose operator T t of T maps L ²(R n , w d x) to itself for all w ∈ A ₁.

7.4.8. Let 1 < p < ∞. Suppose that T is a linear operator that maps L p (v) to itself for all v satisfying v ⁻¹ ∈ A p . Show that the transpose operator T t of T maps L p′(w) to itself for all w satisfying $$w^{-1} \in A_{p'}$$.

7.5 Further Properties of A p Weights

In this section we discuss other properties of A p weights. Many of these properties indicate deep connections with other branches of analysis. We focus attention on three such properties: factorization, extrapolation, and relations of weighted inequalities to vector-valued inequalities.

7.5.1 Factorization of Weights

Recall the simple fact that if w ₁, w ₂ are A ₁ weights, then $$w = w_{1}\,w_{2}^{1-p}$$ is an A p weight (Exercise 7.1.2). The factorization theorem for weights says that the converse of this statement is true. This provides a surprising and striking representation of A p weights.

Theorem 7.5.1.

Suppose that w is an A p weight for some 1 < p < ∞. Then there exist A ₁ weights w ₁ and w ₂ such that

Proof.

Let us fix a p ≥ 2 and w ∈ A p . We define an operator T as follows:

where M is the Hardy–Littlewood maximal operator. We observe that T is well defined and bounded on L p (R n ). This is a consequence of the facts that $$w^{- \frac{1} {p-1} }$$ is an A p′ weight and that M maps $$L^{p'}(w^{- \frac{1} {p-1} })$$ to itself and also L p (w) to itself. Thus the norm of T on L p depends only on the A p characteristic constant of w. Let $$B(w) =\| T\|_{L^{p}\rightarrow L^{p}}$$, the norm of T on L p . Next, we observe that for f, g ≥ 0 in L p (R n ) and λ ≥ 0 we have

(7.5.1)

To see the first assertion, we need only note that for every ball B, the operator

is sublinear as a consequence of Minkowski’s integral inequality, since p − 1 ≥ 1.

We now fix an L p function f ₀ with $$\|f_{0}\|_{L^{p}} = 1$$ and we define a function $$\varphi$$ in $$L^{p}(\mathbf{R}^{n})$$ as the sum of the L p convergent series

(7.5.2)

We define

so that $$w = w_{1}\,w_{2}^{1-p}$$. It remains to show that w ₁, w ₂ are A ₁ weights. Applying T and using (7.5.1), we obtain

that is,

Using that $$\varphi = (w^{-\frac{1} {p} }w_{1})^{ \frac{1} {p-1} } = w^{\frac{1} {p} }w_{2}$$, we obtain

These show that w ₁ and w ₂ are A ₁ weights whose characteristic constants depend on $$[w]_{A_{p}}$$ (and also the dimension n and p). This concludes the case p ≥ 2.

We now turn to the case p < 2. Given a weight w ∈ A p for 1 < p < 2, we consider the weight $$w^{-1/(p-1)}$$, which is in A p′. Since p′ > 2, using the result we obtained, we write $$w^{-1/(p-1)} = v_{1}v_{2}^{1-p'}$$, where v ₁, v ₂ are A ₁ weights. It follows that $$w = v_{1}^{1-p}v_{2}$$, and this completes the asserted factorization of A p weights. □ 

Combining the result just obtained with Theorem 7.2.7, we obtain the following description of A p weights.

Corollary 7.5.2.

Let w be an A p weight for some 1 < p < ∞. Then there exist locally integrable functions f ₁ and f ₂ with

constants $$0 <\varepsilon _{1},\varepsilon _{2} < 1$$ , and a nonnegative function k satisfying k,k ⁻¹ ∈ L ∞ such that

(7.5.3)

7.5.2 Extrapolation from Weighted Estimates on a Single $$L^{p_{0}}$$

Our next topic concerns a striking application of the class of A p weights. It says that an estimate on $$L^{p_{0}}(v)$$ for a single p ₀ and all $$A_{p_{0}}$$ weights v implies a similar L p estimate for all p in (1, ∞). This property is referred to as extrapolation.

Surprisingly the operator T is not needed to be linear or sublinear in the following extrapolation theorem. The only condition required is that T be well defined on $$\bigcup _{1\leq q<\infty }\bigcup _{w\in A_{q}}L^{q}(w)$$. If T happens to be a linear operator, this condition can be relaxed to T being well defined on $$\mathcal{C}_{0}^{\infty }(\mathbf{R}^{n})$$.

Theorem 7.5.3.

Suppose that T is defined on $$\bigcup _{1\leq q<\infty }\bigcup _{w\in A_{q}}L^{q}(w)$$ and takes values in the space of measurable complex-valued functions. Let 1 ≤ p ₀ < ∞ and suppose that there exists a positive increasing function N on [1,∞) such that for all weights v in $$A_{p_{0}}$$ we have

(7.5.4)

Then for any 1 < p < ∞ and for all weights w in A p we have

(7.5.5)

where

and κ ₁ (n,p,p ₀ ) and κ ₂ (n,p,p ₀ ) are constants that depend on n, p, and p ₀ .

Proof.

Let 1 < p < ∞ and w ∈ A p . We define an operator

where M is the Hardy–Littlewood maximal operator. We observe that since w 1−p′ is in A p′, the operator M′ maps L p′(w) to itself; indeed, we have

(7.5.6)

in view of Theorem 7.1.9 and property (4) of Proposition 7.1.5.

We introduce operators M ⁰(f) =  | f | and M k  = M ∘ M ∘⋯ ∘ M, where M is the Hardy–Littlewood maximal function and the composition is taken k times. Likewise, we introduce powers (M′) k of M′ for $$k \in \mathbf{Z}^{+} \cup \{ 0\}$$. The following lemma provides the main tool in the proof of Theorem 7.5.3. Its simple proof uses Theorem 7.1.9 and (7.5.6) and is omitted.

Lemma 7.5.4.

Let 1 < p < ∞ and w ∈ A p . Define operators R and R′

for functions f in L p (w) and also

for functions f in L p′ (w). Then there exist constants C ₁ (n,p) and C ₂ (n,p) that depend on n and p such that

(7.5.7)

(7.5.8)

(7.5.9)

for all functions f in L p (w) and such that

(7.5.10)

(7.5.11)

(7.5.12)

for all functions h in L p′ (w).

We now proceed with the proof of the theorem. It is natural to split the proof into the cases p < p ₀ and p > p ₀.

Case (1): p < p ₀. Assume momentarily that $$R(f)^{- \frac{p_{0}} {(p_{0}/p)'} }$$ is an $$A_{p_{0}}$$ weight. Then we have

where we used Hölder’s inequality with exponents p ₀∕p and (p ₀∕p)′, the hypothesis of the theorem, (7.5.7), and (7.5.8). Thus, we have the estimate

(7.5.13)

and it remains to obtain a bound for the $$A_{p_{0}}$$ characteristic constant of $$R(f)^{- \frac{p_{0}} {(p_{0}/p)'} }$$. In view of (7.5.9), the function R(f) is an A ₁ weight with characteristic constant at most a constant multiple of $$[w]_{A_{p}}^{ \frac{1} {p-1} }$$. Consequently, there is a constant C ₁′ such that

for any cube Q in R n . Thus we have

(7.5.14)

Next we have

(7.5.15)

where we applied Hölder’s inequality with exponents

and we used that

Multiplying (7.5.14) by (7.5.15) and taking the supremum over all cubes Q in R n we deduce that

Combining this estimate with (7.5.13) and using the fact that N is an increasing function, we obtain the validity of (7.5.5) in the case p < p ₀.

Case (2): p > p ₀. In this case we set $$r = p/p_{0} > 1$$. Then we have

(7.5.16)

for some nonnegative function h with L r′(w) norm equal to 1. We define a function

Obviously, we have 0 ≤ h ≤ H and thus

(7.5.17)

noting that

which is valid in view of (7.5.11). Moreover, this argument is based on the hypothesis of the theorem and requires that H w be an $$A_{p_{0}}$$ weight. To see this, we observe that condition (7.5.12) implies that $$H^{r'/p'}w$$ is an A ₁ weight with characteristic constant at most a multiple of $$[w]_{A_{1}}$$. Thus, there is a constant C ₂′ that depends only on n and p such that

for all cubes Q in R n . From this it follows that

where we set $$\kappa _{2}(n,p,p_{0}) = (C_{2}')^{p'/r'}$$. We raise the preceding displayed expression to the power p ₀′ − 1, we average over the cube Q, and then we raise to the power p ₀ − 1. We deduce the estimate

(7.5.18)

where we use the fact that

Note that r′∕p′ ≥ 1, since p ₀ ≥ 1. Using Hölder’s inequality with exponents r′∕p′ and $$(r'/p')^{-1}$$ we obtain that

(7.5.19)

where we used that

Multiplying (7.5.18) by (7.5.19), we deduce the estimate

Inserting this estimate in (7.5.17) we obtain

and combining this with (7.5.16) we conclude that

This proves the required estimate (7.5.5) in the case p > p ₀. □ 

There is a version of Theorem 7.5.3 in which the initial strong type assumption is replaced by a weak type estimate.

Theorem 7.5.5.

Suppose that T is a well defined operator on $$\bigcup _{1<q<\infty }\bigcup _{w\in A_{q}}L^{q}(w)$$ that takes values in the space of measurable complex-valued functions. Fix 1 ≤ p ₀ < ∞ and suppose that there is an increasing function N on [1,∞) such that for all weights v in $$A_{p_{0}}$$ we have

(7.5.20)

Then for any 1 < p < ∞ and for all weights w in A p we have

(7.5.21)

where $$K\big(n,p,p_{0},[w]_{A_{p}}\big)$$ is as in Theorem 7.5.3 .

Proof.

For every fixed λ > 0 we define

The operator T λ is not linear but is well defined on $$\bigcup _{1<q<\infty }\bigcup _{w\in A_{q}}L^{q}(w)$$, since T is well defined on this union. We show that T λ maps $$L^{p_{0}}(v)$$ to $$L^{p_{0}}(v)$$ for every $$v \in A_{p_{0}}$$. Indeed, we have

using the hypothesis on T. Applying Theorem 7.5.3, we obtain that T λ maps L p (w) to itself for all 1 < p < ∞ and all w ∈ A p with a constant independent of λ. Precisely, for any w ∈ A p and any f ∈ L p (w) we have

Since

it follows that T maps L p (w) to L p, ∞ (w) with the asserted norm. □ 

Assuming that the operator T in the preceding theorem is sublinear (or quasi-sublinear), we obtain the following result that contains a stronger conclusion.

Corollary 7.5.6.

Suppose that T is a sublinear operator on $$\bigcup _{1<q<\infty }\bigcup _{w\in A_{q}}L^{q}(w)$$ that takes values in the space of measurable complex-valued functions. Fix 1 ≤ p ₀ < ∞ and suppose that there is an increasing function N on [1,∞) such that for all weights v in $$A_{p_{0}}$$ we have

(7.5.22)

Then for any 1 < p < ∞ and any weight w in A p there is a constant $$K'(n,p,p_{0},[w]_{A_{p}})$$ such that

Proof.

The proof follows from Theorem 7.5.5 and the Marcinkiewicz interpolation theorem. □ 

We end this subsection by observing that the conclusion of the extrapolation Theorem 7.5.3 can be strengthened to yield vector-valued estimates. This strengthening may be achieved by a simple adaptation of the proof discussed.

Corollary 7.5.7.

Suppose that T is defined on $$\bigcup _{1\leq q<\infty }\bigcup _{w\in A_{q}}L^{q}(w)$$ and takes values in the space of all measurable complex-valued functions. Fix 1 ≤ p ₀ < ∞ and suppose that there is an increasing function N on [1,∞) such that for all weights v in $$A_{p_{0}}$$ we have

Then for every 1 < p < ∞ and every weight w ∈ A p we have

for all sequences of functions f j in L p (w), where $$K\big(n,p,p_{0},[w]_{A_{p}}\big)$$ is as in Theorem 7.5.3 .

Proof.

To derive the claimed vector-valued inequality follow the proof of Theorem 7.5.3 replacing the function f by $$(\sum _{j}\vert f_{j}\vert ^{p_{0}})^{ \frac{1} {p_{0}} }$$ and T(f) by $$(\sum _{j}\vert T(f_{j})\vert ^{p_{0}})^{ \frac{1} {p_{0}} }$$. □ 

7.5.3 Weighted Inequalities Versus Vector-Valued Inequalities

We now discuss connections between weighted inequalities and vector-valued inequalities. The next result provides strong evidence that there is a nontrivial connection of this sort. The following is a general theorem saying that any vector-valued inequality is equivalent to some weighted inequality. The proof of the theorem is based on a minimax lemma whose precise formulation and proof can be found in Appendix H.

Theorem 7.5.8.

(a) Let 0 < p < q,r < ∞. Let {T j } j be a sequence of sublinear operators that map L q (μ) to L r (ν), where μ and ν are arbitrary measures. Then the vector-valued inequality

(7.5.23)

holds for all f j ∈ L q (μ) if and only if for every u ≥ 0 in $$L^{ \frac{r} {r-p} }(\nu )$$ there exists U ≥ 0 in $$L^{ \frac{q} {q-p} }(\mu )$$ with

(7.5.24)

(b) Let 0 < q,r < p < ∞. Let {T j } j be as before. Then the vector-valued inequality (7.5.23) holds for all f j ∈ L q (μ) if and only if for every u ≥ 0 in $$L^{ \frac{q} {p-q} }(\mu )$$ there exists U ≥ 0 in $$L^{ \frac{r} {p-r} }(\nu )$$ with

(7.5.25)

Proof.

We begin with part (a). Given f j  ∈ L q (R n , μ), we use (7.5.24) to obtain

which proves (7.5.23) with the same constant C as in (7.5.24). To prove the converse, given a nonnegative $$u \in L^{ \frac{r} {r-p} }(\nu )$$ with $$\|u\|_{L^{ \frac{r}{r-p} }} = 1$$, we define

and

Notice that A and B are convex sets and B is weakly compact. (The sublinearity of each T j is used here.) We define the function Φ on A × B by setting

Then Φ is concave on A and weakly continuous and convex on B. Thus the minimax lemma in Appendix H is applicable. This gives

(7.5.26)

At this point observe that for a fixed $$a =\big (\sum _{j}\vert f_{j}\vert ^{p},\sum _{j}\vert T_{j}(f_{j})\vert ^{p}\big)$$ in A we have

using the hypothesis (7.5.23). It follows that $$\sup _{a\in A}\min _{b\in B}\varPhi (a,b) \leq 0$$ and hence (7.5.26) yields $$\min _{b\in B}\sup _{a\in A}\varPhi (a,b) \leq 0$$. Thus there exists a U ∈ B such that Φ(a, U) ≤ 0 for every a ∈ A. This completes the proof of part (a).

The proof of part (b) is similar. Using the result of Exercise 7.5.1 and (7.5.25), given f j  ∈ L q (R n , μ) we have

To prove the converse direction in part (b), given a fixed u ≥ 0 in $$L^{ \frac{q} {p-q} }(\mu )$$ with $$\big\|u\big\|_{ L^{ \frac{q} {p-q} }} = 1$$, we define A as in part (a) and

We also define the function Φ on A × B by setting

Then Φ is concave on A and weakly continuous and convex on B. Also, using Exercise 7.5.1, for any $$a =\big (\sum _{j}\vert f_{j}\vert ^{p},\sum _{j}\vert T_{j}(f_{j})\vert ^{p}\big)$$ in A, we have

Thus $$\sup _{a\in A}\min _{b\in B}\varPhi (a,b) \leq 0$$. Using (7.5.26), yields $$\min _{b\in B}\sup _{a\in A}\varPhi (a,b) \leq 0$$, and the latter implies the existence of a U in B such that Φ(a, U) ≤ 0 for all a ∈ A. This proves (7.5.25). □ 

Example 7.5.9.

We use the previous theorem to obtain another proof of the vector-valued Hardy–Littlewood maximal inequality in Corollary 5.​6.​5.​ We take T j  = M for all j. For given 1 < p < q < ∞ and u in $$L^{ \frac{q} {q-p} }$$ we set $$s = \frac{q} {q-p}$$ and $$U =\| M\|_{L^{s}\rightarrow L^{s}}^{-1}\,M(u)$$. In view of Exercise 7.1.7 we have

Using Theorem 7.5.8, we obtain

(7.5.27)

whenever 1 < p < q < ∞, an inequality obtained earlier in (5.​6.​25).

It turns out that no specific properties of the Hardy–Littlewood maximal function were used in the preceding inequality, and one could obtain a general result along these lines.

Exercises

7.5.1. Let (X, μ) be a measure space, 0 < s < 1, and f ∈ L s (X, μ). Show that

and that the infimum is attained.

[Hint: Try $$u = c\,\vert f\vert ^{1-s}$$ for a suitable constant c.]

7.5.2. (K. Yabuta) Let w ∈ A p for some 1 < p < ∞ and let f be in L loc p (R n , w d x). Show that f lies in L loc ¹(R n ).

[Hint: Write $$w = w_{1}/w_{2}^{p-1}$$ via Theorem 7.5.1.]

7.5.3. Use the same idea of the proof of Theorem 7.5.1 to prove the following general result: Let μ be a positive measure on a measure space X and let T be a bounded sublinear operator on L p (X, μ) for some 1 ≤ p < ∞. Suppose that T(f) ≥ 0 for all f in L p (X, μ). Prove that for all f ₀ ∈ L p (X, μ), there exists an f ∈ L p (X, μ) such that

(a)

f ₀(x) ≤ f(x) for μ-almost all x ∈ X.

(b)

$$\|f\|_{L^{p}(X)} \leq 2\,\|f_{0}\|_{L^{p}(X)}$$.

(c)

$$T(f)(x) \leq 2\,\|T\|_{L^{p}\rightarrow L^{p}}f(x)$$ for μ-almost all x ∈ X.

[Hint: Try the expression in (7.5.2) starting the sum at j = 0.]

7.5.4. ([100]) Suppose that T is an operator defined on $$\bigcup _{1<q<\infty }\bigcup _{w\in A_{q}}L^{q}(w)$$ that satisfies $$\|T\|_{L^{r}(v)\rightarrow L^{r}(v)} \leq N([v]_{A_{r}})$$ for some increasing function N: [1, ∞) → R ⁺. Without using Theorem 7.5.3 prove that for 1 < q < r and all v ∈ A ₁, T maps L q (v) to L q (v) with constant depending on q, r, n, and $$[v]_{A_{1}}$$.

[Hint: Hölder’s inequality gives that

Then use the fact that the weight $$M(f)^{\frac{r-q} {r-1} }$$ is in A ₁ and Exercise 7.1.2.]

7.5.5. Let T be a sublinear operator defined on $$\bigcup _{2\leq q<\infty }L^{q}$$. Suppose that for all functions f and u we have

Prove that T maps L p (R n ) to itself for all 2 < p < ∞.

[Hint: Use that

and Hölder’s inequality.]

7.5.6. (X. C. Li) Let T be a sublinear operator defined on $$\bigcup _{1<q\leq 2}\bigcup _{w\in A_{q}}L^{q}(w)$$. Suppose that T maps L ²(w) to L ²(w) for all weights w that satisfy w ⁻¹ ∈ A ₁. Prove that T maps L p to itself for all 1 < p < 2.

[Hint: We have

by Hölder’s inequality. Apply the hypothesis to the first term of the product.]

HISTORICAL NOTES

Weighted inequalities can probably be traced back to the beginning of integration, but the A p condition first appeared in a paper of Rosenblum [298] in a somewhat different form. The characterization of A p when n = 1 in terms of the boundedness of the Hardy–Littlewood maximal operator was obtained by Muckenhoupt [260]. The estimate on the norm in (7.1.25) can also be reversed, as shown by Buckley [38]. The simple proof of Theorem 7.1.9 is contained in Lerner’s article [218] and yields both the Muckenhoupt theorem and Buckley’s optimal growth of the norm of the Hardy–Littlewood maximal operator in terms of the A p characteristic constant of the weight. Another proof of this result is given by Christ and Fefferman [61]. Versions of Lemma 7.1.10 for balls were first obtained by Besicovitch [27] and independently by Morse [258]. The particular version of Lemma 7.1.10 that appears in the text is adapted from that in de Guzmán [93]. Another version of this lemma is contained in the book of Mattila [246]. The fact that A ∞ is the union of the A p spaces was independently obtained by Muckenhoupt [261] and Coifman and Fefferman [66]. The latter paper also contains a proof that A p weights satisfy the crucial reverse Hölder condition. This condition first appeared in the work of Gehring [125] in the following context: If F is a quasiconformal homeomorphism from R n into itself, then | det(∇F) | satisfies a reverse Hölder inequality. The characterization of A ₁ weights is due to Coifman and Rochberg [68]. The fact that M(f) δ is in A ∞ when δ < 1 was previously obtained by Córdoba and Fefferman [74]. The different characterizations of A ∞ (Theorem 7.3.3) are implicit in [260] and [66]. Another characterization of A ∞ in terms of the Gurov-Reshetnyak condition $$\sup _{Q} \frac{1} {\vert Q\vert }\int _{Q}\vert f -\mathop{\mbox{ Avg}}_{Q}f\vert \,dx \leq \varepsilon \mathop{\mbox{ Avg}}_{Q}f$$ for f ≥ 0 and $$0 <\varepsilon < 2$$ was obtained by Korenovskyy, Lerner, and Stokolos [201]. The definition of A ∞ using the reverse Jensen inequality herein was obtained as an equivalent characterization of that space by García-Cuerva and Rubio de Francia [122] (p. 405) and independently by Hrusčev [161]. The reverse Hölder condition was extensively studied by Cruz-Uribe and Neugebauer [82].

Weighted inequalities with weights of the form | x |  a for the Hilbert transform were first obtained by Hardy and Littlewood [147] and later by Stein [332] for other singular integrals. The necessity and sufficiency of the A p condition for the boundedness of the Hilbert transform on weighted L p spaces was obtained by Hunt, Muckenhoupt, and Wheeden [167]. Historically, the first result relating A p weights and the Hilbert transform is the Helson-Szegő theorem [149], which says that the Hilbert transform is bounded on L ²(w) if and only if $$\log w = u + Hv$$, where $$u,v \in L^{\infty }(\mathbf{R})$$ and $$\|v\|_{L^{\infty }} < \frac{\pi } {2}$$. The Helson-Szegő condition easily implies the A ₂ condition, but the only known direct proof for the converse gives $$\|v\|_{L^{\infty }} <\pi$$; see Coifman, Jones, and Rubio de Francia [67]. A related result in higher dimensions was obtained by Garnett and Jones [123]. Weighted L p estimates controlling Calderón–Zygmund operators by the Hardy–Littlewood maximal operator were obtained by Coifman [65]. Coifman and Fefferman [66] extended one-dimensional weighted norm inequalities to higher dimensions and also obtained good lambda inequalities for A ∞ weights for more general singular integrals and maximal singular integrals (Theorem 7.4.3). Bagby and Kurtz [19], and later Alvarez and Pérez [4], gave a sharper version of Theorem 7.4.3, by replacing the good lambda inequality by a rearrangement inequality. See also the related work of Lerner [217]. The following relation $$\|M_{d}(f)\|_{L^{p}(w)} \leq C(p,n,[w]_{A_{\infty }})\|M^{\#}(f)\|_{L^{p}(w)}$$ between the dyadic maximal function and the sharp maximal function is valid for any w ∈ A ∞ under the condition $$M(f) \in L^{p_{0}}$$ but also under the weaker assumption that w({ | f |  > t}) < ∞ for every t > 0; see Kurtz [208]. Using that min(M, w) is an A ∞ weight with constant independent of M and Fatou’s lemma, this condition can be relaxed to | { | f |  > t} |  < ∞ for every t > 0. A rearrangement inequality relating f and M # (f) is given in Bagby and Kurtz [18].

The factorization of A p weights was conjectured by Muckenhoupt and proved by Jones [179]. The simple proof given in the text can be found in [67]. Extrapolation of operators (Theorem 7.5.3) is due to Rubio de Francia [300]. An alternative proof of this theorem was given later by García-Cuerva [121]. The value of the constant $$K(n,p,p_{0},[w]_{A_{p}})$$ first appeared in Dragičević, Grafakos, Pereyra, and Petermichl [98]. Another proof with sharp bounds (in terms of the characteristic constant of the weights) was given by Duoandikoetxea [101]. The present treatment of Theorem 7.5.3, based on crucial Lemma 7.5.4, was communicated to the author by J. M. Martell. One may also consult the related work of Cruz-Uribe, Martell, and Pérez [80]. The simple proof of Theorem 7.5.5 was conceived by J. M. Martell and first appeared in the treatment of extrapolation of operators of many variables; see Grafakos and Martell [135]. The idea of extrapolation can be carried to general pairs of functions, see Cruz-Uribe, Martell, and Pérez [78]. Estimates for the distribution function in extrapolation theory were obtained by Carro, Torres, and Soria [58]. The equivalence between vector-valued inequalities and weighted norm inequalities of Theorem 7.5.8 is also due to Rubio de Francia [299]. The difficult direction in this equivalence is obtained using a minimax principle (see Fan [111]). Alternatively, one can use the factorization theory of Maurey [247], which brings an interesting connection with Banach space theory. The book of García-Cuerva and Rubio de Francia [122] provides an excellent reference on this and other topics related to weighted norm inequalities.

A primordial double-weighted norm inequality is the observation of Fefferman and Stein [115] that the maximal function maps L p (M(w)) to L p (w) for nonnegative measurable functions w (Exercise 7.1.7). Sawyer [312] obtained that the condition $$\sup _{Q}\!\big(\int _{Q}v^{1-p'}dx\big)^{-1}\!\!\!\int _{Q}M(v^{1-p'}\chi _{Q})^{p}w\,dx < \infty $$ provides a characterization of all pairs of weights (v, w) for which the Hardy–Littlewood maximal operator M maps L p (v) to L p (w). Simpler proofs of this result were obtained by Cruz-Uribe [77] and Verbitsky [367]. The fact that Sawyer’s condition reduces to the usual A p condition when v = w was shown by Hunt, Kurtz, and Neugebauer [166]. The two-weight problem for singular integrals is more delicate, since they are not necessarily bounded from L p (M(w)) to L p (w). Known results in this direction are that singular integrals map L p (M [p]+1(w)) to L p (w), where M r denotes the rth iterate of the maximal operator. See Wilson [377] (for 1 < p < 2) and Pérez [277] for the remaining p’s. A necessary condition for the boundedness of the Hilbert transform from L p (v) to L p (w) was obtained by Muckenhoupt and Wheeden [262].

For an approach to two-weighted inequalities using Bellman functions, we refer to the article of Nazarov, Treil, and Volberg [266]. The notion of Bellman functions originated in control theory; the article [267] of the previous authors analyzes the connections between optimal control and harmonic analysis. Bellman functions have been used to derive estimates for the norms of classical operators on weighted Lebesgue spaces; for instance, Petermichl [279] showed that for $$w \in A_{2}(\mathbf{R})$$, the norm of the Hilbert transform from $$L^{2}(\mathbf{R},w)$$ to $$L^{2}(\mathbf{R},w)$$ is bounded by a constant times the characteristic constant $$[w]_{A_{2}}$$.

The theory of A p weights in this chapter carries through to the situation in which Lebesgue measure is replaced by a general doubling measure. This theory also has a substantial analogue when the underlying measure is nondoubling but satisfies μ(∂ Q) = 0 for all cubes Q in R n with sides parallel to the axes; see Orobitg and Pérez [272]. A thorough account of weighted Littlewood–Paley theory and exponential-square function integrability is contained in the book of Wilson [378].

The conjecture whether $$\|T\|_{L^{1}(M(w))\rightarrow L^{1,\infty }(w)} < \infty $$ holds for a weight w was disproved by Reguera [287] when T is a Haar multiplier and then by Reguera and Thiele [288] for the Hilbert transform. However, the slightly weaker version of this inequality, in which M(w) is replaced by the Orlicz maximal operator $$M_{L(\log L)^{\varepsilon }}(w)$$, holds for any $$\varepsilon > 0$$ and any Calderón-Zygmund operator T, as shown by Pérez [277]. For A ¹ weights w the aforementioned conjecture would imply $$\|T\|_{L^{1}(w)\rightarrow L^{1,\infty }(w)} \leq C\,[w]_{A_{1}}$$. However, Nazarov, Reznikov, Vasyunin, and Volberg [265] disproved the weaker inequality $$\|T\|_{L^{1}(w)\rightarrow L^{1,\infty }(w)} \leq C\,[w]_{A_{1}}\big(\log (e + [w]_{A_{1}})\big)^{\alpha }$$ for $$\alpha < \frac{1} {5}$$. Lerner, Ombrosi, and Pérez [223] had previously shown that the preceding inequality holds with α = 1 for any Calderón-Zygmund operator T.

Concerning the sharp weighted bound $$\|T\|_{L^{2}(w)\rightarrow L^{2}(w)} \leq c_{T}\,[w]_{A_{2}}$$ for a Calderón-Zygmund operator T we have the work of Petermichl and Volberg [281] which answered a question by Astala, Iwaniecz and Saksman [14] on the regularity of solutions to the Beltrami equation. The proofs of this inequality for the Hilbert and Riesz transforms via the Bellman function technique were obtained soon afterwards by Petermichl [279], [280]. The use of Bellman functions was first avoided in the work of Lacey, Petermichl, and Reguera [210], whose proof recovered the already known cases and used Haar shift operators, the two-weight theory for them of Nazarov, Treil and Volberg [268], and corona decompositions. The simplest proof for these classical operators was obtained by Cruz-Uribe, Martell, and Pérez [79], [81] using a very powerful inequality due to Lerner [219]. The complete proof for a general Calderón-Zygmund operator was given by Hytönen [168]. A simplified proof was provided by Lerner [220], [221]. For other improvements and estimates involving A p and A ∞ constants see the work of Lerner [222], Hytönen and Pérez [170], and Lacey, Hytönen, and Pérez [169].

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Footnotes

1

the dependence on p is relevant only when p < ∞

Appendix A: Gamma and Beta Functions

A.1 A Useful Formula

The following formula is valid:

This is an immediate consequence of the corresponding one-dimensional identity

which is usually proved from its two-dimensional version by switching to polar coordinates:

A.2 Definitions of $$\varGamma (z)$$ and B ( z , w )

For a complex number z with Re  z  > 0 define

$$\varGamma (z)$$ is called the gamma function. It follows from its definition that $$\varGamma (z)$$ is analytic on the right half-plane Re  z  > 0.

Two fundamental properties of the gamma function are that

where z is a complex number with positive real part and $$n \in \mathbf{Z}^{+}$$ . Indeed, integration by parts yields

Since $$\varGamma (1) = 1$$ , the property $$\varGamma (n) = (n - 1)!$$ for $$n \in \mathbf{Z}^{+}$$ follows by induction. Another important fact is that

This follows easily from the identity

Next we define the beta function. Fix z and w complex numbers with positive real parts. We define

We have the following relationship between the gamma and the beta functions:

when z and w have positive real parts.

The proof of this fact is as follows:

A.3 Volume of the Unit Ball and Surface of the Unit Sphere

We denote by v n the volume of the unit ball in $$\mathbf{R}^{n}$$ and by $$\omega _{n-1}$$ the surface area of the unit sphere $$\mathbf{S}^{n-1}$$ . We have the following:

and

The easy proofs are based on the formula in Appendix A.1. We have

by switching to polar coordinates. Now change variables t  =  r ² to obtain that

This proves the formula for the surface area of the unit sphere in $$\mathbf{R}^{n}$$ .

To compute v n , write again using polar coordinates

Here is another way to relate the volume to the surface area. Let B (0,  R ) be the ball in $$\mathbf{R}^{n}$$ of radius R  > 0 centered at the origin. Then the volume of the shell B (0,  R + h ) ∖ B (0,  R ) divided by h tends to the surface area of B (0,  R ) as h  → 0. In other words, the derivative of the volume of B (0,  R ) with respect to the radius R is equal to the surface area of B (0,  R ). Since the volume of B (0,  R ) is $$v_{n}R^{n}$$ , it follows that the surface area of B (0,  R ) is $$n\,v_{n}R^{n-1}$$ . Taking R  = 1, we deduce $$\omega _{n-1} = n\,v_{n}$$ .

A.4 Computation of Integrals Using Gamma Functions

Let $$k_{1},\ldots,k_{n}$$ be nonnegative even integers. The integral

expressed in polar coordinates is equal to

where $$\theta = (\theta _{1},\ldots,\theta _{n})$$ . This leads to the identity

Another classical integral that can be computed using gamma functions is the following:

whenever a and b are complex numbers with $$\mbox{ Re}\,a > -1$$ and $$\mbox{ Re}\,b > -1$$ .

Indeed, change variables $$u = (\sin \varphi )^{2}$$ ; then $$du = 2(\sin \varphi )(\cos \varphi )d\varphi$$ , and the preceding integral becomes

A.5 Meromorphic Extensions of B ( z , w ) and $$\varGamma (z)$$

Using the identity $$\varGamma (z + 1) = z\,\varGamma (z)$$ , we can easily define a meromorphic extension of the gamma function on the whole complex plane starting from its known values on the right half-plane. We give an explicit description of the meromorphic extension of $$\varGamma (z)$$ on the whole plane. First write

and observe that the second integral is an analytic function of z for all $$z \in \mathbf{C}$$ . Write the first integral as

The last integral converges when $$\mbox{ Re}\,z > -N - 1$$ , since the expression inside the curly brackets is O ( t N +1 ) as $$t \rightarrow 0$$ . It follows that the gamma function can be defined to be an analytic function on Re  z  > − N − 1 except at the points z  = − j , $$j = 0,1,\ldots,N$$ , at which it has simple poles with residues $$\frac{(-1)^{j}} {j!}$$ . Since N was arbitrary, it follows that the gamma function has a meromorphic extension on the whole plane.

In view of the identity

the definition of B ( z ,  w ) can be extended to $$\mathbf{C} \times \mathbf{C}$$ . It follows that B ( z ,  w ) is a meromorphic function in each argument.

A.6 Asymptotics of $$\varGamma (x)$$ as $$x \rightarrow \infty $$

We now derive Stirling’s formula :

First change variables $$t = x + sx\sqrt{\frac{2} {x}}$$ to obtain

Setting $$y = \sqrt{\frac{x} {2}},$$ we obtain

To show that the last integral converges to $$\sqrt{\pi }$$ as $$y \rightarrow \infty $$ , we need the following: (1) The fact that

which follows easily by taking logarithms and applying L’Hôpital’s rule twice. (2) The estimate, valid for $$y \geq 1$$ ,

which can be easily checked using calculus. Using these facts, the Lebesgue dominated convergence theorem, the trivial fact that $$\chi _{-y<s<\infty } \rightarrow 1$$ as $$y \rightarrow \infty $$ , and the identity in Appendix A.1, we obtain that

As a consequence of Stirling’s formula, for any t  > 0, we obtain

A.7 Euler’s Limit Formula for the Gamma Function

For n a positive integer and $$\mbox{ Re}\,z > 0$$ we consider the functions

We show that

and we obtain Euler’s limit formula for the gamma function

We write $$\varGamma (z) -\varGamma _{n}(z) = I_{1}(z) + I_{2}(z) + I_{3}(z)$$ , where

Obviously I ₁ ( z ) tends to zero as $$n \rightarrow \infty $$ . For I ₂ and I ₃ we have that 0 ≤  t  <  n , and by the Taylor expansion of the logarithm we obtain

where

It follows that

and thus I ₂ ( z ) tends to zero as $$n \rightarrow \infty $$ . For I ₃ we have t ∕ n  ≤ 1∕2, which implies that

Consequently, for t ∕ n  ≤ 1∕2 we have

Plugging this estimate into I ₃ , we deduce that

which certainly tends to zero as $$n \rightarrow \infty $$ .

Next, n integrations by parts give

This can be written as

Taking limits as $$n \rightarrow \infty $$ , we obtain an infinite product form of Euler’s limit formula ,

where $$\mbox{ Re}\,z > 0$$ and $$\gamma$$ is Euler’s constant

The infinite product converges uniformly on compact subsets of the complex plane that excludes $$z = 0,-1,-2,\ldots$$ , and thus it represents a holomorphic function in this domain. This holomorphic function multiplied by $$\varGamma (z)\,z\,e^{\gamma z}$$ is equal to 1 on $$\mbox{ Re}\,z > 0$$ and by analytic continuation it must be equal to 1 on $$\mathbf{C}\setminus \{0,-1,-2,\ldots \}$$ . But $$\varGamma (z)$$ has simple poles, while the infinite product vanishes to order one, at the nonpositive integers. We conclude that Euler’s limit formula holds for all complex numbers z ; consequently, $$\varGamma (z)$$ has no zeros and $$\varGamma (z)^{-1}$$ is entire.

An immediate consequence of Euler’s limit formula is the identity

which holds for x and y real with $$x\notin \{0,-1,-2,\ldots \}$$ . As a consequence we have that

and also that

where

whenever $$y \in \mathbf{R}$$ and $$x \in \mathbf{R}\setminus \{0,-1,-2,\ldots \}$$ . Before we find a similar estimate for $$x \in \{ 0,-1,-2,\ldots \}$$ we provide a simpler expression for this estimate when x  > 0.

When x  > 0 we have

Thus we conclude that when x  > 0 and $$y \in \mathbf{R}$$ we have

When x  = 0 we write $$\varGamma (iy)iy =\varGamma (1 + iy)$$ and use the preceding inequality to obtain

and more generally for $$x = -N \in \{-1,-2,\ldots \}$$ and $$y \in \mathbf{R}$$ we obtain by induction

A.8 Reflection and Duplication Formulas for the Gamma Function

The reflection formula relates the values of the gamma function of a complex number z and its reflection about the point 1∕2 in the following way:

The duplication formula relates the entire functions $$\varGamma (2z)^{-1}$$ and $$\varGamma (z)^{-1}$$ as follows:

Both of these could be proved using Euler’s limit formula. The reflection formula also uses the identity

while the duplication formula makes use of the fact that

These and other facts related to the gamma function can be found in Olver [ 271 ].

Appendix B: Bessel Functions

B.1 Definition

We survey some basics from the theory of Bessel functions J ν of complex order ν with $$\mbox{ Re}\,\nu > -1/2$$ . We define the Bessel function J ν of order ν by its Poisson representation formula

where $$\mbox{ Re}\,\nu > -1/2$$ and t  ≥ 0. Although this definition is also valid when t is a complex number, for the applications we have in mind, it suffices to consider the case that t is real and nonnegative; in this case J ν ( t ) is also a real number.

B.2 Some Basic Properties

Let us summarize a few properties of Bessel functions. We take t  > 0.

(1) We have the following recurrence formula:

(2) We also have the companion recurrence formula:

(3) J ν ( t ) satisfies the differential equation:

(4) If $$\nu \in \mathbf{Z}^{+}$$ , then we have the following identity, which was taken by Bessel as the definition of J ν for integer ν :

(5) For $$\mbox{ Re}\,\nu > -1/2$$ we have the following identity:

which can also be written as

using the equality $$(2j)! = 2^{2j}\,j!\,(j -\frac{1} {2})(j -\frac{3} {2})\cdots \frac{1} {2} = 2^{2j}\,j!\,\varGamma (j + \frac{1} {2})\,\varGamma (\frac{1} {2})^{-1}$$ . (6) For $$\mbox{ Re}\,\nu > 1/2$$ the identity below is valid:

We first verify property (1). We have

where we integrated by parts and used the fact that $$\varGamma (x + 1) = x\varGamma (x)$$ .

We now prove Property (2) for $$\mbox{ Re}\,\nu > 1/2$$ :

We proceed with the proof of property (3). A calculation using the definition of the Bessel function gives that the left-hand side of (3) is equal to

which in turn is equal to

Property (4) can be derived directly from (1). Define

for $$\nu = 0,1,2,\ldots$$ and t  > 0. We can show easily that $$G_{0} = J_{0}$$ . If we had

for $$\nu \in \mathbf{Z}^{+}$$ , we would immediately conclude that $$G_{\nu } = J_{\nu }$$ for $$\nu \in \mathbf{Z}^{+}$$ . We have

For t real, the identity in (5) can be derived by inserting the expression

for e its in the definition of the Bessel function J ν ( t ) in Appendix B.1. Algebraic manipulations yield

To derive property (6) we first multiply (1) by t ν and (2) by t − ν ; then we use the product rule for differentiation and we add the resulting expressions.

For further identities on Bessel functions, one may consult Watson’s monograph [ 371 ].

B.3 An Interesting Identity

Let $$\mbox{ Re}\,\mu > -\frac{1} {2}$$ , $$\mbox{ Re}\,\nu > -1$$ , and t  > 0. Then the following identity is valid:

To prove this identity we use formula (5) in Appendix B.2. We have

B.4 The Fourier Transform of Surface Measure on $$\mathbf{S}^{n-1}$$

Let $$d\sigma$$ denote surface measure on $$\mathbf{S}^{n-1}$$ for n  ≥ 2. Then the following is true:

To see this, use the result in Appendix D.3 to write

B.5 The Fourier Transform of a Radial Function on $$\mathbf{R}^{n}$$

Let f ( x ) =  f ₀ ( |  x  | ) be a radial function defined on $$\mathbf{R}^{n}$$ , where f ₀ is defined on $$[0,\infty )$$ . Then the Fourier transform of f is given by the formula

To obtain this formula, use polar coordinates to write

As an application we take $$f(x) =\chi _{B(0,1)}(x)$$ , where B (0, 1) is the unit ball in $$\mathbf{R}^{n}$$ . We obtain

in view of the result in Appendix B.3. More generally, for $$\mbox{ Re}\,\lambda > -1$$ , let

Then

using again the identity in Appendix B.3.

B.6 Bessel Functions of Small Arguments

We seek the behavior of J k ( r ) as r  → 0+. We fix $$\nu \in \mathbf{C}$$ with $$\mbox{ Re}\,\nu > -\frac{1} {2}$$ . Then we have the identity

where

and S ν satisfies

To prove this estimate we note that

where we evaluated the last integral using the result in Appendix A.4. Using that $$\vert e^{irt} - 1\vert \leq r\vert t\vert $$ , we deduce the assertion regarding the size of |  S ν ( r ) | .

It follows from these facts and the estimate in Appendix A.7 that for 0 <  r  ≤ 1 and $$\mbox{ Re}\,\nu > -1/2$$ we have

where C ₀ is a constant that depends smoothly on $$\mbox{ Re}\,\nu \in (-1/2,\infty )$$ .

B.7 Bessel Functions of Large Arguments

For r  > 0 and complex numbers ν with Re  ν  > −1∕2 we prove the identity

Fix $$0 <\delta < 1/10 < 10 < R < \infty $$ . We consider the region $$\varOmega _{\delta,R}$$ in the complex plane whose boundary is the set consisting of the interval $$[-1+\delta,1-\delta ]$$ union a quarter circle centered at 1 of radius $$\delta$$ from $$1-\delta$$ to $$1 + i\delta$$ , union the line segments from $$1 + i\delta$$ to 1 + iR , from 1 + iR to − 1 + iR , and from − 1 + iR to $$-1 + i\delta$$ , union a quarter circle centered at − 1 of radius $$\delta$$ from $$-1 + i\delta$$ to $$-1+\delta$$ . This is a simply connected region on the interior of which the holomorphic function (1 − z ² ) has no zeros. Since $$\varOmega _{\delta,R}$$ is contained in the complement of the negative imaginary axis, there is a holomorphic branch of the logarithm such that log( t ) is real, $$\log (-t) =\log \vert t\vert + i\pi$$ , and log( it ) = log |  t  | + iπ ∕2 for t  > 0. Since the function log(1 − z ² ) is well defined and holomorphic in $$\varOmega _{\delta,R}$$ , we may define the holomorphic function

for $$z \in \varOmega _{\delta,R}$$ . Since $$e^{irz}(1 - z^{2})^{\nu -\frac{1} {2} }$$ has no poles in $$\varOmega _{\delta,R}$$ , Cauchy’s theorem yields

where $$E(\delta,R)$$ is the sum of the integrals over the two small quarter-circles of radius $$\delta$$ and the line segment from 1 + iR to − 1 + iR . The first two of these integrals are bounded by constants times $$\delta$$ , the latter by a constant times $$R^{2\mbox{ Re}\,\nu -1}e^{-rR}$$ ; hence $$E(\delta,R) \rightarrow 0$$ as $$\delta \rightarrow 0$$ and $$R \rightarrow \infty $$ . We deduce the identity

Estimating the two integrals on the right by putting absolute values inside and multiplying by the missing factor $$r^{\nu }2^{-\nu }(\varGamma (\nu +\frac{1} {2})\varGamma (\frac{1} {2}))^{-1}$$ , we obtain

since the absolute value of the argument of $$t^{2} \pm 2it$$ is at most $$\pi /2$$ . When $$\mbox{ Re}\,\nu > 1/2$$ , we use the inequality $$(\sqrt{t^{2 } + 4}\,)^{\mbox{ Re}\,\nu -\frac{1} {2} } \leq 2^{\mbox{ Re}\,\nu -\frac{3} {2} }\big(t^{\mbox{ Re}\,\nu -\frac{1} {2} } + 2^{\mbox{ Re}\,\nu -\frac{1} {2} }\big)$$ to get

When $$1/2 \geq \mbox{ Re}\,\nu > -1/2$$ we use that $$\big(\sqrt{t^{2 } + 4}\,\big)^{\mbox{ Re}\,\nu -\frac{1} {2} } \leq 1$$ to deduce that

These estimates yield that for $$\mbox{ Re}\,\nu > -1/2$$ and r  ≥ 1 we have

using the result in Appendix A.7, where C ₁ is a constant that depends smoothly on $$\mbox{ Re}\,\nu$$ on the interval $$(-1/2,\infty )$$ .

B.8 Asymptotics of Bessel Functions

We obtain asymptotics for J ν ( r ) as $$r \rightarrow \infty $$ whenever $$\mbox{ Re}\,\nu > -1/2$$ . We have the following identity for r  > 0:

where $$R_{\nu }$$ is given by

and satisfies $$\vert R_{\nu }(r)\vert \leq C_{\nu }\,r^{-3/2}$$ whenever r  ≥ 1.

To see the validity of this identity we write

Inserting these expressions into the corresponding integrals in the formula proved in Appendix B.7, adding and subtracting 1 from each term $$(1 \pm \tfrac{it} {2} )^{\nu -\frac{1} {2} }$$ , and multiplying by the missing factor $$(r/2)^{\nu }/\varGamma (\nu +\frac{1} {2})\varGamma (\frac{1} {2})$$ , we obtain the claimed identity

It remains to estimate R ν ( r ). We begin by noting that for a ,  b real with a  > −1 we have the pair of inequalities

The first inequality is proved by splitting into real and imaginary parts and applying the mean value theorem in the real part. Taking $$\nu -\frac{1} {2} = a + ib$$ , y  =  t ∕2, and inserting these estimates into the integrals appearing in R ν , we obtain

It follows that for all r  > 0 we have

using that $$e^{-t} \leq e^{-t/2}e^{-r}$$ for t  ≥ 2 r . We conclude that for r  ≥ 1 and $$\mbox{ Re}\,\nu > -1/2$$ we have

via the result in Appendix A.7, where C ₂ is a constant that depends smoothly on $$\mbox{ Re}\,\nu$$ on the interval $$(-1/2,\infty )$$ .

B.9 Bessel Functions of general complex indices

We now discuss how to extend J ν ( t ) to complex values of ν when t  > 0. Identity (5)

of Appendix B.2 expresses J ν ( t ) as a power series when ν is a complex number with $$\mbox{ Re}\,\nu > -1/2$$ . But this power series converges absolutely for all complex values of ν by the ratio test. Indeed, for j sufficiently large we have

which tends to zero as $$j \rightarrow \infty $$ , thus the ratio test applies.

Therefore the power series defines an entire function of ν . Since this entire function coincides with J ν on $$(-\frac{1} {2},\infty ) \times \mathbf{R}$$ , by analytic continuation, J ν ( t ) has an entire extension for all t  > 0.

We obtain an integral representation of J z ( t ) when $$-1/2 \geq \mbox{ Re}\,z > -3/2$$ . Fix t  > 0. Then we write

where the last step follows by the change of variables u  = 1 − s ² . By the mean value theorem, there is a constant c in (0, 1) such that

where the second identity is due to the fact that the integral $$\int _{0}^{1}u^{z-\frac{1} {2} }\,du$$ is equal to $$(z + \frac{1} {2})^{-1}$$ when $$\mbox{ Re}\,z > -1/2$$ and thus when divided by $$\varGamma (z + \frac{1} {2})$$ can be analytically extended to an entire function. It follows from these calculations that

We now estimate J z ( t ) when $$-3/2 < \mbox{ Re}\,z \leq -1/2$$ . Given q  ∈ (0, 1) we write

where

for some c satisfying 0 <  c  <  q .

Suppose that t  > 2. Then we pick q  = 1∕ t . We have

for some constant C ₁ depending on $$\mbox{ Re}\,z$$ . The second term obviously satisfies

In $$J_{z}^{3}(t;1/t)$$ we write $$\frac{\cos (t\sqrt{1-u}\,)} {\sqrt{1-u}} = \frac{-2} {t} \frac{d} {du}\big[\sin (t\sqrt{1 - u})\big]$$ and integrate by parts to obtain the estimate

when $$-1/2 \geq \mbox{ Re}\,z > -3/2$$ . Thus for these $$z$$ ’s and t  > 2 we have

When 0 <  t  ≤ 2 we choose q  = 1∕2. Then each of $$\vert J_{z}^{k}(t;1/2)\vert $$ , k  = 1, 2, 3 is bounded by a multiple of $$t^{\mbox{ Re}\,z}$$ and we obtain the estimate

when $$-1/2 \geq \mbox{ Re}\,z > -3/2$$ and 0 <  t  ≤ 2.

In the special case $$\mbox{ Re}\,z = -1/2$$ or z  = −1∕2 + iθ , $$\theta \in \mathbf{R}$$ , we deduce

for some constant C independent of all parameters; see Appendix A.7 for the last inequality.

Appendix C: Rademacher Functions

C.1 Definition of the Rademacher Functions

The Rademacher functions are defined on [0, 1] as follows: r ₀ ( t ) = 1; r ₁ ( t ) = 1 for 0 ≤  t  ≤ 1∕2 and r ₁ ( t ) = −1 for 1∕2 <  t  ≤ 1; r ₂ ( t ) = 1 for 0 ≤  t  ≤ 1∕4, r ₂ ( t ) = −1 for 1∕4 <  t  ≤ 1∕2, r ₂ ( t ) = 1 for 1∕2 <  t  ≤ 3∕4, and r ₂ ( t ) = −1 for 3∕4 <  t  ≤ 1; and so on. According to this definition, we have that $$r_{j}(t) = \mbox{ sgn}(\sin (2^{j}\pi t))$$ for $$j = 0,1,2,\ldots$$ . It is easy to check that the r j ’s are mutually independent random variables on [0, 1]. This means that for all functions f j we have

To see the validity of this identity, we write its right-hand side as

and we observe that there is a one-to-one and onto correspondence between subsets S of $$\{1,2,\ldots,n\}$$ and intervals $$I_{k} =\big [ \tfrac{k} {2^{n}}, \tfrac{k+1} {2^{n}} \big]$$ , $$k = 0,1,\ldots,2^{n} - 1$$ , such that the restriction of the function $$\prod _{j=1}^{n}f_{j}(r_{j}(t))$$ on I k is equal to

It follows that the last of the three equal displayed expressions is

C.2 Khintchine’s Inequalities

The following property of the Rademacher functions is of fundamental importance and with far-reaching consequences in analysis:

For any $$0 < p < \infty $$ and for any real-valued square summable sequences { a j } and { b j } we have

for some constants $$0 < A_{p},B_{p} < \infty $$ that depend only on p .

These inequalities reflect the orthogonality of the Rademacher functions in L p (especially when p ≠ 2). Khintchine [ 193 ] was the first to prove a special form of this inequality, and he used it to estimate the asymptotic behavior of certain random walks. Later this inequality was systematically studied almost simultaneously by Littlewood [ 226 ] and by Paley and Zygmund [ 274 ], who proved the more general form stated previously. The foregoing inequalities are usually referred to by Khintchine’s name.

C.3 Derivation of Khintchine’s Inequalities

Both assertions in Appendix C.2 can be derived from an exponentially decaying distributional inequality for the function

when a j , b j are square summable real numbers.

We first obtain a distributional inequality for the above function F under the following three assumptions:

(a)

The sequence { b j } is identically zero.

(b)

All but finitely many terms of the sequence { a j } are zero.

(c)

The sequence { a j } satisfies $$(\sum _{j}\vert a_{j}\vert ^{2})^{1/2} = 1$$ .

Let ρ  > 0. Under assumptions (a), (b), and (c), independence gives

where we used the inequality $$\frac{1} {2}(e^{x} + e^{-x}) \leq e^{\frac{1} {2} x^{2} }$$ for all real x , which can be checked using power series expansions. Since the same argument is also valid for $$-\sum a_{j}r_{j}(t)$$ , we obtain that

From this it follows that

and hence we obtain the distributional inequality

by picking $$\rho =\alpha$$ . The $$L^{p}$$ norm of F can now be computed easily. Formula ( 1.​1.​6 ) gives

We have now proved that

under assumptions (a), (b), and (c).

We now dispose of assumptions (a), (b), and (c). Assumption (b) can be easily eliminated by a limiting argument and (c) by a scaling argument. To dispose of assumption (a), let a j and b j be real numbers. We have

Let us now set $$A_{p} = 2\big(p\varGamma (p/2)\big)^{1/p}$$ when p  > 2. Since we have the trivial estimate $$\big\|F\big\|_{L^{p}} \leq \big\| F\big\|_{L^{2}}$$ when 0 <  p  ≤ 2, we obtain the required inequality $$\big\|F\big\|_{L^{p}} \leq A_{p}\big\|F\big\|_{L^{2}}$$ with

Using Sterling’s formula in Appendix A.6, we see that A p is asymptotic to $$\sqrt{p}$$ as $$p \rightarrow \infty $$ .

We now discuss the converse inequality $$B_{p}\big\|F\big\|_{L^{2}} \leq \big\| F\big\|_{L^{p}}$$ . It is clear that $$\big\|F\big\|_{L^{2}} \leq \big\| F\big\|_{L^{p}}$$ when p  ≥ 2 and we may therefore take B p  = 1 for p  ≥ 2. Let us now consider the case 0 <  p  < 2. Pick an s such that $$2 < s < \infty $$ . Find a 0 <  θ  < 1 such that

Then

It follows that

We have now proved the inequality $$B_{p}\big\|F\big\|_{L^{2}} \leq \big\| F\big\|_{L^{p}}$$ with

Observe that the function $$s \rightarrow A_{s}^{-\big( \frac{1} {p}-\frac{1} {2} \big)/\big(\frac{1} {2} -\frac{1} {s}\big)}$$ tends to zero as s  → 2+ and as $$s \rightarrow \infty $$ . Hence it must attain its maximum for some s  =  s ( p ) in the interval $$(2,\infty )$$ . We see that $$B_{p} \geq 16 \cdot 256^{-1/p}$$ when p  < 2 by taking s  = 4.

It is worthwhile to mention that the best possible values of the constants A p and B p in Khintchine’s inequality are known when b j  = 0. In this case Szarek [ 353 ] showed that the best possible value of B ₁ is $$1/\sqrt{2}$$ , and later Haagerup [ 141 ] found that when b j  = 0 the best possible values of A p and B p are the numbers

and

where $$p_{0} = 1.84742\ldots$$ is the unique solution of the equation $$2\varGamma (\frac{p+1} {2} ) = \sqrt{\pi }$$ in the interval (1, 2).

C.4 Khintchine’s Inequalities for Weak Type Spaces

We note that the following weak type estimates are valid:

for all $$0 < p < \infty $$ .

Indeed, the upper estimate is a simple consequence of the fact that L p is a subspace of $$L^{p,\infty }$$ . For the converse inequality we use the fact that $$L^{p,\infty }([0,1])$$ is contained in L p ∕2 ([0, 1]) and we have (see Exercise 1.1.11)

Since any Lorentz space $$L^{p,q}([0,1])$$ can be sandwiched between L 2 p ([0, 1]) and L p ∕2 ([0, 1]), similar inequalities hold for all Lorentz spaces L p ,  q ([0, 1]), $$0 < p < \infty $$ , $$0 < q \leq \infty $$ .

C.5 Extension to Several Variables

We first extend the inequality on the right in Appendix C.2 to several variables. For a positive integer n we let

for t j  ∈ [0, 1], where $$c_{j_{1},\ldots,j_{n}}$$ is a sequence of complex numbers and F n is a function defined on [0, 1] n .

For any $$0 < p < \infty $$ and for any complex-valued square summable sequence of n variables $$\{c_{j_{1},\ldots,j_{n}}\}_{j_{1},\ldots,j_{n}}$$ , we have the following inequalities for F n :

where $$A_{p},B_{p}$$ are the constants in Appendix C.2. The norms are over [0, 1] n .

The case n  = 2 is indicative of the general case. For p  ≥ 2 we have

where we used Minkowski’s integral inequality (with exponent p ∕2 ≥ 1) in the second inequality and the result in the case n  = 1 twice.

The case p  < 2 follows trivially from Hölder’s inequality with constant A p  = 1. The reverse inequalities follow exactly as in the case of one variable. Replacing A p by $$A_{p}^{n}$$ in the argument, giving the reverse inequality in the case n  = 1, we obtain the constant $$B_{p}^{n}$$ .

Likewise one may extend the weak type inequalities of Appendix C.3 in several variables.

Appendix D: Spherical Coordinates

D.1 Spherical Coordinate Formula

Switching integration from spherical coordinates to Cartesian is achieved via the following identity:

where

and $$0 \leq \varphi _{1},\ldots,\varphi _{n-2} \leq \pi$$ , $$0 \leq \varphi _{n-1} =\theta < 2\pi$$ ,

and

is the Jacobian of the transformation.

D.2 A Useful Change of Variables Formula

The following formula is useful in computing integrals over the sphere $$\mathbf{S}^{n-1}$$ when n  ≥ 2. Let f be a function defined on $$\mathbf{S}^{n-1}$$ . Then we have

To prove this formula, let $$\varphi ' = (\varphi _{2},\ldots,\varphi _{n-1})$$ and

Using the change of variables in Appendix D.1 we express

as the iterated integral

and we can realize the expression inside the square brackets as

Consequently,

and the change of variables

yields

Rescaling the sphere S n −2 to $$\sqrt{R^{2 } - s^{2}}\,\mathbf{S}^{n-2}$$ yields the claimed identity.

D.3 Computation of an Integral over the Sphere

Let K be a function on the line. We use the result in Appendix D.2 to show that for n  ≥ 2 we have

when $$x \in \mathbf{R}^{n}\setminus \{0\}$$ . Let x ′ =  x ∕ |  x  | and pick a matrix A  ∈  O ( n ) such that Ae ₁  =  x ′, where $$e_{1} = (1,0,\ldots,0)$$ . We have

where $$\omega _{n-2} = 2\pi ^{\frac{n-1} {2} }\varGamma \big(\frac{n-1} {2}\big)^{-1}$$ is the surface area of S n −2 .

For example, we have

and the integral converges only when $$\mbox{ Re}\,\alpha < 1$$ .

D.4 The Computation of Another Integral over the Sphere

We compute the following integral for n  ≥ 2:

where $$e_{1} = (1,0,\ldots,0)$$ . Applying the formula in Appendix D.2, we obtain

which converges exactly when $$\mbox{ Re}\,\alpha < n - 1$$ .

D.5 Integration over a General Surface

Suppose that S is a hypersurface in R n of the form S  = { ( u ,  Φ ( u )):   u  ∈  D }, where D is an open subset of R n −1 and Φ is a continuously differentiable mapping from D to $$\mathbf{R}$$ . Let $$\sigma$$ be the canonical surface measure on S . If g is a function on S , then we have

Specializing to the sphere, we obtain

D.6 The Stereographic Projection

Define a map $$\varPi: \mathbf{R}^{n} \rightarrow \mathbf{S}^{n}$$ by the formula

It is easy to see that Π is a one-to-one map from R n onto the sphere S n minus the north pole $$e_{n+1} = (0,\ldots,0,1)$$ . Its inverse is given by the formula

The Jacobian of the map is verified to be $$J_{\varPi }(x) =\big ( \frac{2} {1+\vert x\vert ^{2}} \big)^{-n}\,,$$ and the following change of variables formulas are valid:

and

where

Another interesting formula about the stereographic projection Π is

Appendix E: Some Trigonometric Identities and Inequalities

The following inequalities are valid for t real:

The following sum to product formulas are valid:

The following identities are also easily proved:

Appendix F: Summation by Parts

Let $$\{a_{k}\}_{k=0}^{\infty }$$ , $$\{b_{k}\}_{k=0}^{\infty }$$ be two sequences of complex numbers. Then for N  ≥ 1 we have

where

More generally we have

whenever 0 ≤  M  ≤  N , where A ₋₁  = 0 and

for k  ≥ 0.

Appendix G: Basic Functional Analysis

A quasi-norm is a nonnegative functional $$\|\,\cdot \,\|$$ on a vector space X that satisfies $$\|x + y\|_{X} \leq K(\|x\|_{X} +\| y\|_{X})$$ for some K  ≥ 0 and all x ,  y  ∈  X and also $$\|\lambda x\|_{X} = \vert \lambda \vert \|x\|_{X}$$ for all scalars $$\lambda$$ . When K  = 1, the quasi-norm is called a norm. A quasi-Banach space is a quasi-normed space that is complete with respect to the topology generated by the quasi-norm. The proofs of the following theorems can be found in several books including Albiac and Kalton [ 1 ], Kalton Peck and Roberts [ 188 ], and Rudin [ 306 ].

The Hahn–Banach theorem. Let X be a normed vector space (over the real or complex numbers), let Y be a subspace of X , and let P be a positively homogeneous subadditive ¹ functional on X . Then for every linear functional $$\varLambda$$ on Y that satisfies

for all y  ∈  Y , there is a linear functional $$\varLambda ^{{\prime}}$$ on X such that

In particular, every bounded linear functional on a subspace has an extension on the entire space with the same norm.

Banach–Alaoglou theorem. Let X be a normed vector space and let X ∗ be the space of all bounded linear functionals on X . Then the closed unit ball of X ∗ is compact in the weak ∗ topology.

A special case is the sequential version of this theorem, which asserts that the closed unit ball of the dual space of a separable normed vector space is sequentially compact in the weak ∗ topology. Indeed, the weak ∗ topology on the closed unit ball of the dual of a separable space is metrizable, and thus compactness and sequential compactness are equivalent.

Open mapping theorem. Suppose that X and Y are quasi-Banach spaces and T is a bounded surjective linear map from X onto Y . Then T maps open sets to open sets, i.e., it is an open mapping. Moreover, if T is injective, then there exists a constant $$K < \infty $$ such that for all x  ∈  X we have

Closed graph theorem. Suppose that X and Y are quasi-Banach spaces and T is a linear map from X to Y whose graph is a closed set, i.e., whenever x k ,  x  ∈  X and $$(x_{k},T(x_{k}))\mapsto (x,y)$$ in X × Y for some y  ∈  Y , then T ( x ) =  y . Then T is a bounded linear map from X to Y .

Uniform boundedness principle. Suppose that X is a quasi-Banach space, Y is a quasi-normed space and $$(T_{\alpha })_{\alpha \in I}$$ is a family of bounded linear maps from X to Y such that for all x  ∈  X there exists a $$C_{x} < \infty $$ such that

Then there exists a constant $$K < \infty $$ such that

Appendix H: The Minimax Lemma

Minimax type results are used in the theory of games and have their origin in the work of Von Neumann [ 369 ]. Much of the theory in this subject is based on convex analysis techniques. For instance, this is the case with the next proposition, which is needed in the “difficult” inequality in the proof of the minimax lemma. We refer to Fan [ 111 ] for a general account of minimax results. The following exposition is based on the simple presentation in Appendix A2 of [ 122 ].

Minimax Lemma.  Let A, B be convex subsets of certain vector spaces. Assume that a topology is defined in B for which it is a compact Hausdorff space and assume that there is a function $$\varPhi: A \times B \rightarrow \mathbf{R}\bigcup \{ + \infty \}$$ that satisfies the following: (a) Φ( . ,b) is a concave function on A for each b ∈ B, (b) Φ(a, . ) is a convex function on B for each a ∈ A,(c) Φ(a, . ) is lower semicontinuous on B for each a ∈ A. Then the following identity holds:

To prove the lemma we need the following proposition:

Proposition.  Let B be a convex compact subset of a vector space and suppose that $$g_{j}: B \rightarrow \mathbf{R}\bigcup \{ + \infty \}$$ , j = 1,2,…,n, are convex and lower semicontinuous functions. If

then there exist nonnegative numbers $$\lambda _{1},\lambda _{2},\ldots,\lambda _{n}$$ such that

Proof. We first consider the case n  = 2. Define subsets of B

If B ₁  =  ∅ , we take λ ₁  = 1 and λ ₂  = 0, and we similarly deal with the case B ₂  =  ∅ . If B ₁ and B ₂ are nonempty, then they are closed and thus compact. The hypothesis of the proposition implies that $$g_{2}(b) > 0 \geq g_{1}(b)$$ for all b  ∈  B ₁ . Therefore, the function $$-g_{1}(b)/g_{2}(b)$$ is well defined and upper semicontinuous on B ₁ and thus attains its maximum. The same is true for $$-g_{2}(b)/g_{1}(b)$$ defined on B ₂ . We set

We need to find λ  > 0 such that $$\lambda g_{1}(b) + g_{2}(b) > 0$$ for all b  ∈  B . This is clearly satisfied if $$b\not\in B_{1}\bigcup B_{2}$$ , while for $$b_{1} \in B_{1}$$ and $$b_{2} \in B_{2}$$ we have

Therefore, it suffices to find a λ  > 0 such that 1 − λ μ ₁  > 0 and λ − μ ₂  > 0. Such a λ exists if and only if $$\mu _{1}\mu _{2} < 1$$ . To prove that $$\mu _{1}\mu _{2} < 1$$ , we can assume that μ ₁ ≠ 0 and μ ₂ ≠ 0. Then we take $$b_{1} \in B_{1}$$ and $$b_{2} \in B_{2}$$ , for which the maxima μ ₁ and μ ₂ are attained, respectively. Then we have

But $$g_{1}(b_{1}) < 0 < g_{1}(b_{2})$$ ; thus taking $$b_{\theta } = \theta b_{1} + (1 -\theta )b_{2}$$ for some θ in (0, 1), we have

Considering the same convex combination of the last displayed equations and using this identity, we obtain that

The hypothesis of the proposition implies that $$g_{2}(b_{\theta }) > 0$$ and the convexity of g ₂ :

Since $$g_{2}(b_{1}) > 0$$ , we must have $$\mu _{1}\mu _{2}g_{2}(b_{1}) < g_{2}(b_{1})$$ , which gives $$\mu _{1}\mu _{2} < 1$$ . This proves the required claim and completes the case n  = 2.

We now use induction to prove the proposition for arbitrary n . Assume that the result has been proved for n − 1 functions. Consider the subset of B

If B n  =  ∅ , we choose $$\lambda _{1} =\lambda _{2} = \cdots =\lambda _{n-1} = 0$$ and λ n  = 1. If B n is not empty, then it is compact and convex and we can restrict $$g_{1},g_{2},\ldots,g_{n-1}$$ to B n . Using the induction hypothesis, we can find $$\lambda _{1},\lambda _{2},\ldots,\lambda _{n-1} \geq 0$$ such that

for all b  ∈  B n . Then g ₀ and g n are convex lower semicontinuous functions on B , and $$\max (g_{0}(b),g_{n}(b)) > 0$$ for all b  ∈  B . Using the case n  = 2, which was first proved, we can find $$\lambda _{0},\lambda _{n} \geq 0$$ such that for all b  ∈  B we have

This establishes the case of n functions and concludes the proof of the induction and hence of the proposition. □ 

We now turn to the proof of the minimax lemma.

Proof. The fact that the left-hand side in the required conclusion of the minimax lemma is at least as big as the right-hand side is obvious. We can therefore concentrate on the converse inequality. In doing this we may assume that the right-hand side is finite. Without loss of generality we can subtract a finite constant from Φ ( a ,  b ), and so we can also assume that

Then, by hypothesis (c) of the minimax lemma, the subsets

of B are closed and nonempty, and we show that they satisfy the finite intersection property. Indeed, suppose that

for some $$a_{1},a_{2},\ldots,a_{n} \in A$$ . We write $$g_{j}(b) =\varPhi (a_{j},b)$$ , j  = 1, 2,  … ,  n , and we observe that the conditions of the previous proposition are satisfied. Therefore we can find $$\lambda _{1},\lambda _{2},\ldots,\lambda _{n} \geq 0$$ such that for all b  ∈  B we have

For simplicity we normalize the λ j ’s by setting $$\lambda _{1} +\lambda _{2} + \cdots +\lambda _{n} = 1$$ . If we set $$a_{0} =\lambda _{1}a_{1} +\lambda _{2}a_{2} + \cdots +\lambda _{n}a_{n}$$ , the concavity hypothesis (a) gives

for all b  ∈  B , contradicting the fact that $$\sup _{a\in A}\min _{b\in B}\varPhi (a,b) = 0$$ . Therefore, the family of closed subsets $$\{B_{a}\}_{a\in A}$$ of B satisfies the finite intersection property. The compactness of B now implies $$\bigcap _{a\in A}B_{a}\neq \varnothing $$ . Take $$b_{0} \in \bigcap _{a\in A}B_{a}$$ . Then $$\varPhi (a,b_{0}) \leq 0$$ for every a  ∈  A , and therefore

as required. □ 

Appendix I: Taylor’s and Mean Value Theorem in Several Variables

I.1 Mutlivariable Taylor’s Theorem

For a multiindex $$\alpha = (\alpha _{1},\ldots,\alpha _{n}) \in (\mathbf{Z}^{+} \cup \{ 0\})^{n}$$ , we denote by $$\vert \alpha \vert =\alpha _{1} + \cdots +\alpha _{n}$$ its size, we define $$\alpha ! =\alpha _{1}!\cdots \alpha _{n}!$$ its factorial, and we set

where $$h = (h_{1},\ldots,h_{n})$$ ; here 0 ⁰  = 1.

Let $$k \in \mathbf{Z}^{+} \cup \{ 0\}$$ . Suppose a real-valued $$\mathcal{C}^{k+1}$$ function f is defined on an open convex subset $$\varOmega$$ of $$\mathbf{R}^{n}$$ . Suppose that x  ∈  Ω and $$x + h \in \varOmega$$ . Then we have the Taylor expansion formula

where the remainder R ( h ,  x ,  k ) can be expressed either in Langrange’s mean value form

for some c  ∈ (0, 1), or in integral form

I.2 The Mean value Theorem

Suppose that f is as above and k  = 0. Then for given $$x,y \in \varOmega$$ we have

for some c  ∈ (0, 1). This is a special case of Taylor’s formula when k  = 0.

Appendix J: The Whitney Decomposition of Open Sets in $$\mathbf{R}^{n}$$

J.1 Decomposition of Open Sets

An arbitrary open set in $$\mathbf{R}^{n}$$ can be decomposed as a union of disjoint cubes whose lengths are proportional to their distance from the boundary of the open set. See, for instance, Figure J.1 when the open set is the unit disk in R ² . For a given cube Q in $$\mathbf{R}^{n}$$ , we denote by ℓ ( Q ) its length.

Proposition. Let $$\varOmega$$ be an open nonempty proper subset of $$\mathbf{R}^{n}$$ . Then there exists a family of closed dyadic cubes $$\{Q_{j}\}_{j}$$ (called the Whitney cubes of $$\varOmega$$ ) such that (a) $$\bigcup _{j}Q_{j} =\varOmega$$ and the Q j ’s have disjoint interiors.(b) $$\sqrt{n}\,\ell(Q_{j}) \leq \text{dist }\!(Q_{j},\varOmega ^{c}) \leq 4\sqrt{n}\,\ell(Q_{j})$$ . Thus $$10\sqrt{n}\,Q_{j}$$ meets $$\varOmega ^{c}$$ . (c) If the boundaries of two cubes Q j and Q k touch, then

(d) For a given Q j there exist at most $$12^{n} - 4^{n}$$ cubes Q k that touch it.(e) Let $$0 <\varepsilon < 1/4$$ . If $$Q_{j}^{{\ast}}$$ has the same center as Q j and $$\ell(Q_{j}^{{\ast}}) = (1+\varepsilon )\ell(Q_{j})$$ then

Proof. Let $$\mathcal{D}_{k}$$ be the collection of all dyadic cubes of the form

where $$m_{j} \in \mathbf{Z}$$ . Observe that each cube in $$\mathcal{D}_{k}$$ gives rise to 2 n cubes in $$\mathcal{D}_{k+1}$$ by bisecting each side.

Write the set $$\varOmega$$ as the union of the sets

over all $$k \in \mathbf{Z}$$ . Let $$\mathcal{F}'$$ be the set of all cubes $$Q$$ in $$\mathcal{D}_{k}$$ for some $$k \in \mathbf{Z}$$ such that $$Q \cap \varOmega _{k}\neq \varnothing $$ . We show that the collection $$\mathcal{F}'$$ satisfies property (b). Let $$Q \in \mathcal{F}'$$ and pick $$x \in \varOmega _{k} \cap Q$$ for some $$k \in \mathbf{Z}$$ . Observe that

which proves (b).

Next we observe that

Indeed, every $$Q$$ in $$\mathcal{F}'$$ is contained in $$\varOmega$$ (since it has positive distance from its complement) and every $$x \in \varOmega$$ lies in some $$\varOmega _{k}$$ and in some dyadic cube in $$\mathcal{D}_{k}$$ .

Fig. J.1

The Whitney decomposition of the unit disk.

The problem is that the cubes in the collection $$\mathcal{F}'$$ may not be disjoint. We have to refine the collection $$\mathcal{F}'$$ by eliminating those cubes that are contained in some other cubes in the collection. Recall that two dyadic cubes have disjoint interiors or else one contains the other. For every cube Q in $$\mathcal{F}'$$ we can therefore consider the unique maximal cube Q max in $$\mathcal{F}'$$ that contains it. Two different such maximal cubes must have disjoint interiors by maximality. Now set $$\mathcal{F} =\{ Q^{\mbox{ max}}:\,\, Q \in \mathcal{F}'\}$$ .

The collection of cubes $$\{Q_{j}\}_{j} = \mathcal{F}$$ clearly satisfies (a) and (b), and we now turn our attention to the proof of (c). Observe that if Q j and Q k in $$\mathcal{F}$$ touch then

which proves (c). To prove (d), note that any cube Q in $$\mathcal{D}_{k}$$ is touched by exactly 3 n − 1 other cubes in $$\mathcal{D}_{k}$$ . But each cube Q in $$\mathcal{D}_{k}$$ can contain at most 4 n cubes of $$\mathcal{F}$$ of length at least one-quarter of the length of Q . This fact combined with (c) yields (d). To prove (e), notice that each $$Q_{j}^{{\ast}}$$ is contained in $$\varOmega$$ by part (b). If $$x \in \varOmega$$ , then $$x \in Q_{k_{0}}$$ for some k ₀ . If Q j does not touch $$Q_{k_{0}}$$ , then Q j ∗ does not touch $$Q_{k_{0}}$$ as well. Consequently, the given $$x \in \varOmega$$ may lie only in $$Q_{k}^{{\ast}}$$ for these cubes Q k that touch $$Q_{k_{0}}$$ and there are $$12^{n} - 4^{n} + 1$$ such cubes including $$Q_{k_{0}}$$ . □ 

J.2 Partition of Unity adapted to Whitney cubes

Let us fix an $$\varepsilon$$ such that $$0 <\varepsilon < 1/4$$ . For each cube Q we denote by Q ∗ the cube with the same center as Q and side length $$(1+\varepsilon )\ell(Q)$$ , where ℓ ( Q ) is the side length of Q . Let us fix a nonnegative smooth function ϕ that is equal to 1 on the unit cube $$Q_{0} = [-1/2,1/2]^{n}$$ and equal to zero outside $$Q_{0}^{{\ast}}$$ .

Let $$\{Q_{k}\}_{k}$$ be the family of Whitney cubes of $$\varOmega$$ . We denote by c k the center of Q k and by ℓ k its side length. Since $$\text{dist}(Q_{k},\varOmega ^{c}) \geq \sqrt{n}\,\ell_{k}$$ , and $$\text{dist}((Q_{k}^{{\ast}})^{c},Q_{k}) \leq \varepsilon \sqrt{n}\,\ell_{k}$$ , it follows that

hence $$Q_{k}^{{\ast}}$$ is contained in $$\varOmega$$ . Since the union of Q k is $$\varOmega$$ , then the union of $$Q_{k}^{{\ast}}$$ is also  $$\varOmega$$ .

For each k we define

and we set

We notice that Φ is smooth and that Φ  ≥ 1 on $$\varOmega$$ . Then we define

Obviously $$\varphi _{k}$$ are supported in $$Q_{k}^{{\ast}}$$ , the union of $$Q_{k}^{{\ast}}$$ is $$\varOmega$$ , and we have

We would like to control $$\vert \partial ^{\alpha }\varphi _{k}(x)\vert $$ in terms of $$\ell_{k}$$ . To achieve this we use the Leibniz rule of differentiation. For a fixed k we have that

and obviously $$\big\vert \partial ^{\alpha -\beta }\phi _{k}(x)\big\vert \leq \ell_{k}^{-\vert \alpha \vert +\vert \beta \vert }$$ . A simple inductive argument shows that

for some constant C , and since for a given $$x \in \varOmega$$ , Φ ( x ) is the sum of at most 12 n functions with nonzero values, it follows that

when $$x \in Q_{k}^{{\ast}}$$ and thus

for $$x \in Q_{k}^{{\ast}}$$ . We conclude that for every multiindex $$\alpha$$ there is a constant $$C_{\alpha,n}$$ such that

More on Whitney decompositions can be found in the article of Whitney [ 373 ] and the books of Stein [ 338 ], Krantz and Parks [ 204 ].

Glossary

A is a subset of B (also denoted by $$A \subseteq B$$ )

A is a proper subset of B

A  ⊃  B

B is a proper subset of A

A c

the complement of a set A

χ E

the characteristic function of the set E

d f

the distribution function of a function f

f ∗

the decreasing rearrangement of a function f

f n ↑ f

f n increases monotonically to a function f

the set of all integers

the set of all positive integers $$\{1,2,3,\ldots \}$$

the n -fold product of the integers

the set of real numbers

the set of positive real numbers

the Euclidean n -space

Q

the set of rationals

Q n

the set of n -tuples with rational coordinates

the set of complex numbers

the n -fold product of complex numbers

the unit circle identified with the interval [0, 1]

the n -dimensional torus [0, 1] n

 |  x  | 

$$\sqrt{\vert x_{1 } \vert ^{2 } +\ldots +\vert x_{n } \vert ^{2}}$$ when $$x = (x_{1},\ldots,x_{n}) \in \mathbf{R}^{n}$$

the unit sphere $$\{x \in \mathbf{R}^{n}:\,\, \vert x\vert = 1\}$$

e j

the vector $$(0,\ldots,0,1,0,\ldots,0)$$ with 1 in the j th entry and 0 elsewhere

log t

the logarithm with base e of t  > 0

log a t

the logarithm with base a of t  > 0 (1 ≠ a  > 0)

log ⁺ t

max(0, log t ) for t  > 0

[ t ]

the integer part of the real number t

x ⋅  y

the quantity $$\sum _{j=1}^{n}x_{j}y_{j}$$ when $$x = (x_{1},\ldots,x_{n})$$ and $$y = (y_{1},\ldots,y_{n})$$

B ( x ,  R )

the ball of radius R centered at x in $$\mathbf{R}^{n}$$

ω n −1

the surface area of the unit sphere $$\mathbf{S}^{n-1}$$

the volume of the unit ball $$\{x \in \mathbf{R}^{n}:\,\, \vert x\vert < 1\}$$

 |  A  | 

the Lebesgue measure of the set $$A \subseteq \mathbf{R}^{n}$$

dx

Lebesgue measure

Avg B   f

the average $$\frac{1} {\vert B\vert }\int _{B}f(x)\,dx$$ of f over the set B

the real inner product $$\int _{\mathbf{R}^{n}}f(x)g(x)\,dx$$

the complex inner product $$\int _{\mathbf{R}^{n}}f(x)\overline{g(x)}\,dx$$

the action of a distribution u on a function f

p ′

the number p ∕( p − 1), whenever $$0 < p\neq 1 < \infty $$

1′

the number $$\infty $$

the number 1

f  =  O ( g )

means |  f ( x ) | ≤  M  |  g ( x ) | for some M for x near x ₀

f  =  o ( g )

means |  f ( x ) |  |  g ( x ) |  ⁻¹  → 0 as x  →  x ₀

A t

the transpose of the matrix A

A ∗

the conjugate transpose of a complex matrix A

A ⁻¹

the inverse of the matrix A

O ( n )

the space of real matrices satisfying $$A^{-1} = A^{t}$$

the norm of the (bounded) operator T :  X  →  Y

A  ≈  B

means that there exists a c  > 0 such that $$c^{-1} \le \frac{B}{A} \le c$$

indicates the size $$\vert \alpha _{1}\vert +\ldots +\vert \alpha _{n}\vert $$ of a multi-index $$\alpha = (\alpha _{1},\ldots,\alpha _{n})$$

the m th partial derivative of $$f(x_{1},\ldots,x_{n})$$ with respect to x j

the space of functions f with $$\partial ^{\alpha }f$$ continuous for all $$\vert \alpha \vert \leq k$$

the space of continuous functions with compact support

the space of continuous functions that vanish at infinity

the space of smooth functions with compact support

the space of smooth functions with compact support

the space of Schwartz functions

the space of Schwartz functions $$\varphi$$ with the property $$\int _{\mathbf{R}^{n}}x^{\gamma }\varphi (x)\,dx = 0$$ for all multi-indices $$\gamma$$ .

the space of smooth functions $$\bigcup _{k=1}^{\infty }\mathcal{C}^{k}$$

the space of distributions on $$\mathbf{R}^{n}$$

the space of tempered distributions on $$\mathbf{R}^{n}$$

the space of distributions with compact support on $$\mathbf{R}^{n}$$

the set of all complex-valued polynomials of n real variables

the space of tempered distributions on $$\mathbf{R}^{n}$$ modulo polynomials

the side length of a cube Q in $$\mathbf{R}^{n}$$

the boundary of a cube $$Q$$ in $$\mathbf{R}^{n}$$

the Lebesgue space over the measure space $$(X,\mu )$$

the space $$L^{p}(\mathbf{R}^{n},\vert \cdot \vert )$$

L p ,  q ( X ,  μ )

the Lorentz space over the measure space ( X ,  μ )

the space of functions that lie in $$L^{p}(K)$$ for any compact set K in $$\mathbf{R}^{n}$$

 |  dμ  | 

the total variation of a finite Borel measure μ on $$\mathbf{R}^{n}$$

the space of all finite Borel measures on $$\mathbf{R}^{n}$$

the space of $$L^{p}$$ Fourier multipliers, $$1 \leq p \leq \infty $$

the space of translation-invariant operators that map $$L^{p}(\mathbf{R}^{n})$$ to $$L^{q}(\mathbf{R}^{n})$$

$$\int _{\mathbf{R}^{n}}\vert d\mu \vert $$ the norm of a finite Borel measure $$\mu$$ on $$\mathbf{R}^{n}$$

the centered Hardy–Littlewood maximal operator with respect to balls

M

the uncentered Hardy–Littlewood maximal operator with respect to balls

the centered Hardy–Littlewood maximal operator with respect to cubes

M c

the uncentered Hardy–Littlewood maximal operator with respect to cubes

the centered maximal operator with respect to a measure μ

M μ

the uncentered maximal operator with respect to a measure μ

M s

the strong maximal operator

M d

the dyadic maximal operator

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Index

$${\mathcal{C}}^{k}$$ functions on the torus

σ -finite

absolutely summable Fourier series

adjoint of an operator

admissible growth

almost everywhere convergence

almost orthogonality

analytic family of operators

Aoki–Rolewicz theorem

A ₁ condition

A p condition

necessity of

approximate identity

asymptotics

of Bessel function

asymptotics of gamma function

atom

in a measure space

bad function

Banach–Alaoglou theorem

Banach-valued extension of a linear operator

Banach-valued extension of an operator

Banach-valued measurable function

Banach-valued singular integral

band limited function

Bernoulli polynomials

Bernstein’s inequality

Bernstein’s theorem

Bessel function

asymptotics

large arguments

small arguments

beta function

beta integral identity

Boas and Bochner inequality

Bochner integral

Bochner–Riesz means

Bochner–Riesz operator

Borel measures

bounded variation

B V , functions of bounded variation

Calderón–Zygmund operator

Calderón–Zygmund decomposition

Calderón–Zygmund decomposition on L q

Calderón–Zygmund operator

maximal

cancellation condition

for a kernel

Carleson operator

Cauchy sequence in measure

centered Hardy–Littlewood maximal function

centered maximal function with respect to cubes

Cesàro means

characteristic constant of a pair of weights

Characteristic constant of a weight

A p

characteristic constant of a weight

A ₁

characterization

of A ₁ weights

of A ∞ weights

Chebyshev’s inequality

circular Dirichlet kernel

circular means

circular partial sum

class ( A p ,  A p )

class of A ₁ weights

class of A p weights

closed

under translations

closed graph theorem

commutes with translations

compactly supported distribution

complete orthonormal system

completeness

of Lorentz spaces

completeness of L p , p  < 1

conditional expectation

cone multiplier

conjugate function

conjugate harmonic

conjugate harmonic functions

conjugate Poisson kernel

continuously differentiable function

of order N

continuously differentiable functions

on the torus

convergence

in $$\mathcal{C}_{0}^{\infty }$$

in $$\mathcal{S}$$

in L p

in $${\mathcal{C}}^{\infty }$$

in measure

in weak L p

convex sequence

convolution

Cotlar’s inequality

countably simple

countably simple function

countably subadditive operator

covering lemma

critical Bochner–Riesz index

critical point

Darboux’s equation

de la Vallée Poussin kernel

decay of Fourier coefficients

decomposition of open sets

decreasing rearrangement

of a simple function

degree of a trigonometric polynomial

deLeeuw’s theorem

derivative

of a distribution

of a function (partial)

differentiation

of approximate identities

theory

dilation

L ¹ dilation

of a function

of a tempered distribution

Dini condition

Dini’s theorem

Dirac mass

directional Hilbert transform

Dirichlet kernel

Dirichlet problem

on the sphere

distribution

homogeneous

of lattice points

supported at a point

tempered

with compact support

distribution function

of a simple function

distributional derivative

divergence

of Bochner–Riesz means at the critical index

of the Fourier series of a continuous function

of the Fourier series of an L ¹ function

doubling condition

on a measure

doubling measure

doubly truncated kernel

doubly truncated singular integrals

duals of Lorentz spaces

duBois Reymond’s theorem

duplication formula

for the gamma function

dyadic cube

dyadic decomposition

dyadic interval

dyadic martingale difference operator

dyadic martingale square function

dyadic maximal function

dyadic spherical maximal function

eigenvalues

of the Fourier transform

equidistributed

equidistributed sequence

essential infimum

Euler’s constant

Euler’s limit formula for the gamma function

infinite product form

expectation

conditional

extrapolation

extrapolation theorem

factorization of A p weights

Fejér kernel

Fejér means

Fejér’s theorem

finite Borel measure

finitely simple function

Fourier coefficient

of a measure

Fourier inversion

on L ¹

on L ²

Fourier multiplier

Fourier series

Fourier transform

of a radial function

of a Schwartz function

of surface measure

on L ¹

on L ²

on L p , 1 <  p  < 2

properties of

Fréchet space

frame

tight

Fresnel integral

fundamental solution

fundamental theorem of algebra

g -function

gamma function

asymptotics

duplication formula

meromorphic extension

generalized Cauchy-Riemann equations

good function

good lambda inequality

consequences of

gradient

gradient condition

for a kernel

Green’s identity

Hölder condition

Hölder’s inequality

for weak spaces

Hörmander’s condition

Hörmander–Mihlin multiplier theorem

Haar function

Haar measure

Haar wavelet

Hadamard’s three lines lemma

Hahn–Banach theorem

Hardy’s inequalities

Hardy–Littlewood maximal function

centered

uncentered

harmonic distribution

harmonic function

Hausdorff–Young

Hausdorff–Young inequality

heat equation

heat kernel

Heisenberg group

Hilbert transform

maximal

maximal directional

truncated

Hirschman’s lemma

homogeneous distribution

homogeneous Lipschitz functions

homogeneous maximal singular integrals

homogeneous singular integrals

improved integrability

of A p weights

inductive limit topology

infinitely differentiable function

inhomogeneous Lipschitz functions

inner product

complex

real

inner regular measure

interpolation

Banach-valued Marcinkiewicz theorem

Banach-valued Riesz–Thorin theorem

for analytic families of operators

Marcinkiewicz theorem

off-diagonal Marcinkiewicz theorem

Riesz–Thorin theorem

Stein’s theorem

with change of measure

inverse Fourier transform

isoperimetric inequality

Jensen’s inequality

Jones’s factorization theorem

Khintchine’s inequalities

for weak type spaces

Kolmogorov’s inequality

Kolmogorov’s theorem

Kronecker’s lemma

lacunary Carleson–Hunt theorem

lacunary sequence

Laplace’s equation

Laplacian

lattice points

Lebesgue constants

Lebesgue differentiation theorem

left Haar measure

left maximal function

Leibniz’s rule

linear operator

Liouville’s theorem

Lipschitz condition

for a kernel

Lipschitz space

homogeneous

inhomogeneous

Littlewood–Paley operator

Littlewood–Paley theorem

localization principle

locally compact topological group

locally finite measure

locally integrable functions

logconvexity of weak L ¹

Lorentz spaces

M. Riesz’s theorem

majorant

radial decreasing

Marcinkiewicz and Zygmund theorem

Marcinkiewicz function

Marcinkiewicz interpolation theorem

Marcinkiewicz multiplier theorem

on R n

on R

maximal Calderón–Zygmund operator

maximal function

centered with respect to cubes

dyadic

dyadic spherical

Hardy–Littlewood centered

Hardy–Littlewood uncentered

left

right

strong

uncentered with respect to cubes

with respect to a general measure

maximal Hilbert transform

maximal singular integral

doubly truncated

maximal singular integrals with even kernels

maximal truncated singular integral

measurable set

method of rotations

metrizability

of Lorentz space L p ,  q

of weak L p

Mihlin-Hörmander multiplier theorem

minimally supported frequency wavelet

minimax lemma

Minkowski’s inequality

integral form

Minkowski’s integral inequality

Muckenhoupt characteristic constant

Muckenhoupt characteristic constant of a weight

multi-index

multiplier

on the torus

multiplier theorems

necessity of A p condition

nonatomic measure space

nonsmooth Littlewood–Paley theorem

normability

p -normability of weak L p for p  < 1

of Lorentz space L p ,  q

of Lorentz spaces

of weak L p for p  > 1

normability of weak L ¹ (lack of)

off-diagonal Marcinkiewicz interpolation theorem

open mapping theorem

operator

commuting with translations

of strong type ( p ,  q )

of weak type ( p ,  q )

orthonormal set

orthonormal system

complete

oscillation of a function

oscillatory integral

overshoot

Parseval’s relation

partial derivative

partition of unity

phase

Plancherel’s identity

pointwise convergence of Fourier series

Poisson kernel

Poisson kernel for the sphere

Poisson representation formula

of Bessel functions

Poisson summation formula

polarization

positive operator

power weights

principal value integral

principle of localization

quasi-linear operator

quasi-subadditive operator

Rademacher functions

radial decreasing majorant

radial function

real analytic function

reflection

of a function

of a tempered distribution

reflection formula

for the gamma function

regulated function

restricted weak type

restricted weak type (1, 1)

restricted weak type ( p ,  q ) with constant C

reverse Hölder condition

of order q

reverse Hölder property of A p weights

R H q ( μ )

Riemann zeta function

Riemann’s principle of localization

Riemann–Lebesgue lemma

Riesz product

Riesz projection

Riesz transform

Riesz–Thorin interpolation theorem

right maximal function

right Haar measure

rotations method

rough singular integrals

Rubio de Francia’s extrapolation theorem

sampling theorem

Schwartz function

Schwartz seminorm

self-adjoint operator

self-transpose operator

Sidon set

simple function

sine integral function

singular integrals with even kernels

size condition

for a kernel

smooth bump

smooth function

smooth function with compact support

smoothly truncated maximal singular integral

smoothness condition

for a kernel

space

L ∞

L p

L p ,  ∞

L p ,  q

spectrum of the Fourier transform

spherical maximal function

spherical average

spherical coordinates

spherical Dirichlet kernel

spherical partial sum

square Dirichlet kernel

square function

dyadic martingale

square function of Littlewood–Paley

square partial sum

standard kernel

Stein’s interpolation theorem

stereographic projection

Stirling’s formula

stopping-time

stopping-time argument

strong maximal function

strong type ( p ,  q )

sublinear operator

summation by parts

support of a distribution

surface area of the unit sphere S n −1

Taylor formula

tempered distribution

test function

tight frame

tiling of R n

topological group

torus

total order of differentiation

transference of maximal multipliers

transference of multipliers

translation

of a function

of a tempered distribution

translation operator

translation-invariant operator

transpose of an operator

trigonometric monomial

trigonometric polynomial

truncated Hilbert transform

truncated maximal singular integral

truncated singular integral

uncentered Hardy–Littlewood maximal function

uncentered maximal function with respect to a general measure

uncentered maximal function with respect to cubes

uncertainty principle

uniform boundedness principle

unitary matrix

Vandermonde determinant

variation of a function

vector-valued

extrapolation theorem

Hardy–Littlewood maximal inequality

inequalities

vector-valued extension of a linear operator

vector-valued inequalities

vector-valued Littlewood–Paley theorem

vector-valued singular integral

volume of the unit ball in R n

wave equation

wavelet

of minimally supported frequency

wavelet transform

weak L p

weak type (1, 1)

weak type ( p ,  q )

Weierstrass approximation theorem

for trigonometric polynomials

Weierstrass’s theorem

weight

of class A ₁

of class A ∞

of class A p

weighted estimates

for singular integral operators

Weyl’s theorem

Whitney decomposition

Young’s covering lemma

Young’s inequality

for weak type spaces

Footnotes

1

this means that P ( x ) ≥ 0 for all x  ∈  X , $$P(\lambda x) =\lambda P(x)$$ for all $$\lambda > 0$$ and all x  ∈  X , and that P ( x + z ) ≤  P ( x ) + P ( z ) for all x ,  z  ∈  X .