Discrete Analogues in Harmonic Analysis: Bourgain, Stein, and Beyond

592 Pages·2022·3.59 MB·English

by  Ben Krause

224
GRADUATE STUDIES
IN MATHEMATICS
Discrete Analogues
in Harmonic Analysis
Bourgain, Stein, and Beyond
Ben Krause
10.1090/gsm/224
Discrete Analogues
in Harmonic Analysis
Bourgain, Stein, and Beyond

224
GRADUATE STUDIES
IN MATHEMATICS
Discrete Analogues
in Harmonic Analysis
Bourgain, Stein, and Beyond
Ben Krause
EDITORIAL COMMITTEE
Matthew Baker
Marco Gualtieri
Gigliola Staffilani (Chair)
Jeff A. Viaclovsky
Rachel Ward
2020 Mathematics Subject Classification. Primary 11-XX, 37-XX, 42-XX.
For additional informationand updates on this book, visit
www.ams.org/bookpages/gsm-224
Library of Congress Cataloging-in-Publication Data
Names: Krause,Ben,1988-author.
Title: Discreteanaloguesinharmonicanalysis: Bourgain,Stein,andbeyond/BenKrause.
Description: Providence,RhodeIsland: AmericanMathematicalSociety,2022. |Series: Graduate
studiesinmathematics,1065-7339;224|Includesbibliographicalreferencesandindex.
Identifiers: LCCN2022024899|ISBN9781470468576(hardcover)|ISBN9781470471743(paper-
back)|9781470471750(ebook)
Subjects: LCSH: Harmonic analysis. | AMS: Number theory. | Dynamical systems and ergodic
theory. |HarmonicanalysisonEuclideanspaces.
Classification: LCCQA403.K642022|DDC515/.2433–dc23/eng20220826
LCrecordavailableathttps://lccn.loc.gov/2022024899
DOI:https://doi.org/10.1090/gsm/224
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Contents
List of Symbols xi
Foreword xxi
Acknowledgements xxv
Introduction 1
§1. Discrete Analogues in Harmonic Analysis in Action 1
§2. A Brief History of Discrete Analogues in Harmonic Analysis 3
§3. Where We’re Going 8
§4. Part One: Harmonic Analytic Preliminaries 8
§5. Part Two: Discrete Analogues in Harmonic Analysis: Radon
Transforms I 9
§6. Part Three: Discrete Analogues in Harmonic Analysis: Radon
Transforms II 10
§7. PartFour: DiscreteAnaloguesinHarmonicAnalysis: Maximally
Modulated Singular Integrals 11
§8. Part Five: Discrete Analogues in Harmonic Analysis: An
Introduction to Multilinear Theory 12
§9. Part Six: Conclusion and Appendices 12
Part 1. Harmonic Analytic Preliminaries
Chapter 1. Tools 17
§1. Exploiting Invariance 18
§2. Interpolation of Lp-Spaces 23
vii
viii Contents
§3. The Hardy-Littlewood Maximal Function 33
§4. Continuous Operators on Infinite Dimensional Vector Spaces 47
§5. The Fourier Transform on (cid:2)2(Z) 50
§6. The Euclidean Fourier Transform 56
Chapter 2. On Oscillation and Convergence 73
Chapter 3. The Linear Theory 81
§1. The Pointwise Ergodic Theorem 81
§2. Birkhoff’s Theorem 82
§3. Introduction to Variation 91
§4. The Proof of L´epingle’s Inequality 126
Part 2. Discrete Analogues in Harmonic Analysis: Radon
Transforms, I
Chapter 4. Bourgain’s Maximal Functions on (cid:2)2(Z) 137
§1. Number Theoretic Approximations 142
§2. The Multi-Frequency Maximal Theory: Preliminaries 156
§3. Controling a Maximal Function on L2 161
§4. Proving the Multifrequency Maximal Theory 168
§5. Oscillation and Convergence 178
Chapter 5. Random Pointwise Ergodic Theory 183
§1. Probabilistic Preliminaries 183
§2. The Lp(X)-Theory, 1 < p < ∞ 193
§3. L1(X)-Considerations 196
Chapter 6. An Application to Discrete Ramsey Theory 219
§1. Sarko¨zy’s Theorem 220
§2. A “Pinned” Sark¨ozy Theorem 226
Chapter 7. Bourgain’s (cid:2)2(Z)-Argument, Revisited 235
§1. The Interpolative Approach 235
§2. Exploiting “Superorthogonality,” Preliminary Version 249
Part 3. Discrete Analogues in Harmonic Analysis: Radon
Transforms, II
Chapter 8. Ionescu-Wainger Theory 257
§1. The Introduction of Ionescu-Wainger Theory 257
Contents ix
§2. Applications 263
§3. The Proof of the Variational Estimate 265
Chapter 9. Establishing Ionescu-Wainger Theory 273
§1. The Proof of the Ionescu-Wainger Construction 275
§2. Proof of the Decoupling Estimate 290
Chapter 10. The Spherical Maximal Function 301
§1. The Euclidean Theory 301
§2. The Discrete Theory 305
§3. Number Theoretic Approximations 306
§4. The Restricted Weak-Type Argument 315
Chapter 11. The Lacunary Spherical Maximal Function 319
§1. Euclidean Lacunary Averaging Operators 320
§2. The Lacunary Discrete Spherical Maximal Function 329
Chapter 12. Discrete Improving Inequalities 339
§1. Continuous Improving Inequalities: Spherical Averages 340
§2. Discrete Improving Inequalities 341
§3. Connections to Fractional Integration 354
Part 4. Discrete Analogues in Harmonic Analysis: Maximally
Modulated Singular Integrals
Chapter 13. Monomial “Carleson” Operators 363
§1. Introduction 363
§2. TT∗ Preliminaries and the Continuous Theory 367
§3. A Top-Down Sketch of the Argument 375
§4. Most Modulation Parameters are Safe: A TT∗ Argument 376
§5. Approximations 382
§6. Most Weyl Sums are Safe 389
§7. The Multi-Frequency Theory: Completing the Proof 393
Chapter 14. Maximally Modulated Singular Integrals: A Theorem of
Stein and Wainger 403
§1. Introduction 403
§2. A Reveiw of Stein-Wainger 407
§3. Exponential Sums and Sublevel Estimates 412
§4. The Discrete Stein-Wainger Operator 417