This timely book explores certain modern topics and connections at the interface of harmonic analysis, ergodic theory, number theory, and additive combinatorics. The main ideas were pioneered by Bourgain and Stein, motivated by questions involving averages over polynomial sequences, but the subject has grown significantly over the last 30 years, through the work of many researchers, and has steadily become one of the most dynamic areas of modern harmonic analysis. The author has succeeded admirably in choosing and presenting a large number of ideas in a mostly self-contained and exciting monograph that reflects his interesting personal perspective and expertise into these topics. --Alexandru Ionescu, Princeton University Discrete harmonic analysis is a rapidly developing field of mathematics that fuses together classical Fourier analysis, probability theory, ergodic theory, analytic number theory, and additive combinatorics in new and interesting ways. While one can find good treatments of each of these individual ingredients from other sources, to my knowledge this is the first text that treats the subject of discrete harmonic analysis holistically. The presentation is highly accessible and suitable for students with an introductory graduate knowledge of analysis, with many of the basic techniques explained first in simple contexts and with informal intuitions before being applied to more complicated problems; it will be a useful resource for practitioners in this field of all levels. --Terence Tao, University of California, Los Angeles
Author(s): Ben Krause
Series: Graduate Studies in Mathematics, 224
Publisher: AMS
Year: 2022
Language: English
Commentary: decrypted from 3EE323843D2D9A68783DA95A8B79CA62 source file
Pages: 563
City: Providence
List of Symbols
Asymptotic Notation
Miscellaneous Symbols
Functions and Operators in the Euclidean Setting
Functions and Operators in the Discrete Setting
Fourier Multipliers in the Discrete Setting
Foreword
Acknowledgements
Introduction
1. Discrete Analogues in Harmonic Analysis in Action
2. A Brief History of Discrete Analogues in Harmonic Analysis
3. Where We’re Going
4. Part One: Harmonic Analytic Preliminaries
5. Part Two: Discrete Analogues in Harmonic Analysis: Radon Transforms I
6. Part Three: Discrete Analogues in Harmonic Analysis: Radon Transforms II
7. Part Four: Discrete Analogues in Harmonic Analysis: Maximally Modulated Singular Integrals
8. Part Five: Discrete Analogues in Harmonic Analysis: An Introduction to Multilinear Theory
9. Part Six: Conclusion and Appendices
Part 1. Harmonic Analytic Preliminaries
Chapter 1. Tools
1. Exploiting Invariance
2. Interpolation of ?^{?}-Spaces
3. The Hardy-Littlewood Maximal Function
4. Continuous Operators on Infinite Dimensional Vector Spaces
5. The Fourier Transform on ℓ²(ℤ)
6. The Euclidean Fourier Transform
Chapter 2. On Oscillation and Convergence
Chapter 3. The Linear Theory
1. The Pointwise Ergodic Theorem
2. Birkhoff’s Theorem
3. Introduction to Variation
4. The Proof of Lépingle’s Inequality
Part 2. Discrete Analogues in Harmonic Analysis: Radon Transforms, I
Chapter 4. Bourgain’s Maximal Functions on ℓ²(ℤ)
1. Number Theoretic Approximations
2. The Multi-Frequency Maximal Theory: Preliminaries
3. Controling a Maximal Function on ?²
4. Proving the Multifrequency Maximal Theory
5. Oscillation and Convergence
Chapter 5. Random Pointwise Ergodic Theory
1. Probabilistic Preliminaries
2. The ?^{?}(?)-Theory, 1
3. ?¹(?)-Considerations
Chapter 6. An Application to Discrete Ramsey Theory
1. Sarközy’s Theorem
2. A “Pinned” Sarközy Theorem
Chapter 7. Bourgain’s ℓ²(ℤ)-Argument, Revisited
1. The Interpolative Approach
2. Exploiting “Superorthogonality,” Preliminary Version
Part 3. Discrete Analogues in Harmonic Analysis: Radon Transforms, II
Chapter 8. Ionescu-Wainger Theory
1. The Introduction of Ionescu-Wainger Theory
2. Applications
3. The Proof of the Variational Estimate
Chapter 9. Establishing Ionescu-Wainger Theory
1. The Proof of the Ionescu-Wainger Construction
2. Proof of the Decoupling Estimate
Chapter 10. The Spherical Maximal Function
1. The Euclidean Theory
2. The Discrete Theory
3. Number Theoretic Approximations
4. The Restricted Weak-Type Argument
Chapter 11. The Lacunary Spherical Maximal Function
1. Euclidean Lacunary Averaging Operators
2. The Lacunary Discrete Spherical Maximal Function
Chapter 12. Discrete Improving Inequalities
1. Continuous Improving Inequalities: Spherical Averages
2. Discrete Improving Inequalities
3. Connections to Fractional Integration
Part 4. Discrete Analogues in Harmonic Analysis: Maximally Modulated Singular Integrals
Chapter 13. Monomial “Carleson” Operators
1. Introduction
2. ??* Preliminaries and the Continuous Theory
3. A Top-Down Sketch of the Argument
4. Most Modulation Parameters are Safe: A ??* Argument
5. Approximations
6. Most Weyl Sums are Safe
7. The Multi-Frequency Theory: Completing the Proof
Chapter 14. Maximally Modulated Singular Integrals: A Theorem of Stein and Wainger
1. Introduction
2. A Reveiw of Stein-Wainger
3. Exponential Sums and Sublevel Estimates
4. The Discrete Stein-Wainger Operator
5. Approximations
6. Analytic Estimates
Part 5. Discrete Analogues in Harmonic Analysis: an Introduction to Multilinear Theory
Chapter 15. Bilinear Considerations
1. Back to Ergodic Theory
2. The Bilinear Theory: Proof Ideas
3. The Pinned Incidence Estimate
Chapter 16. Arithmetic Sobolev Estimates: Examples
1. Arithmetic Sobolev Estimates on ℤ/?ℤ
2. Sobolev Estimates for the Continuous Bilinear Averages
3. The ?^{∞}-Inverse Theorem
4. The Improving Argument on ℝ
Part 6. Conclusion and Appendices
Chapter 17. Further Directions
Remembering my Collaboration with Stein and Bourgain –M. Mirek
Appendix A. Introduction to Additive Combinatorics
1. Roth’s Theorem
2. Introduction to Gowers Norms
3. Higher Order Uniformity Norms
Appendix B. Oscillatory Integrals and Exponential Sums
1. The One-Dimensional Theory
2. Higher Dimensions
3. Applications: Weyl Sum Estimates
Bibliography
Index
