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Chapter 1. Preliminaries
1.1. Notation and Terminology
1.2. Some Results for Real Analysis
1.3. Some Results from Complex Analysis
1.4. Green’s Theorem
Chapter 2. Pointwise Convergence Almost Everywhere
2.1. The Magic of Maximal Functions
2.2. Distribution Functions, Weak-L1, and Interpolation
2.3. The Hardy–Littlewood Maximal Inequality
2.4. Differentiation and Convolution
2.5. Comparison of Measures
2.6. The Maximal and Birkhoff Ergodic Theorems
2.7. Applications of the Ergodic Theorems
2.8. Bonus Section: More Applications of the Ergodic Theorems
2.9. Bonus Section: Subadditive Ergodic Theorem and Lyapunov Behavior
2.10. Martingale Inequalities and Convergence
2.11. The Christ–Kiselev Maximal Inequality and Pointwise Convergence of Fourier Transforms
Chapter 3. Harmonic and Subharmonic Functions
3.1. Harmonic Functions
3.2. Subharmonic Functions
3.3. Bonus Section: The Eremenko–Sodin Proof of Picard’s Theorem
3.4. Perron’s Method, Barriers, and Solution of the Dirichlet Problem
3.5. Spherical Harmonics
3.6. Potential Theory
3.7. Bonus Section: Polynomials and Potential Theory
3.8. Harmonic Function Theory of Riemann Surfaces
Chapter 4. Bonus Chapter: Phase Space Analysis
4.1. The Uncertainty Principle
4.2. The Wavefront Sets and Products of Distributions
4.3. Microlocal Analysis: A First Glimpse
4.4. Coherent States
4.5. Gabor Lattices
4.6. Wavelets
Chapter 5. Hp Spaces and Boundary Values of Analytic Functions on the Unit Disk
5.1. Basic Properties of Hp
5.2. H2
5.3. First Factorization (Riesz) and Hp
5.4. Caratheodory Functions, h1, and the Herglotz Representation
5.5. Boundary Value Measures
5.6. Second Factorization (Inner and Outer Functions)
5.7. Conjugate Functions and M. Riesz’s Theorem
5.8. Homogeneous Spaces and Convergence of Fourier Series
5.9. Boundary Values of Analytic Functions in the Upper Half-Plane
5.10. Beurling’s Theorem
5.11. Hp-Duality and BMO
5.12. Cotlar’s Theorem on Ergodic Hilbert Transforms
Chapter 6. Bonus Chapter: More Inequalities
6.1. Lorentz Spaces and Real Interpolation
6.2. Hardy-Littlewood–Sobolev and Stein–Weiss Inequalities
6.3. Sobolev Spaces; Sobolev and Rellich–Kondrachov Embedding Theorems
6.4. The Calderon–Zygmund Method
6.5. Pseudodifferential Operators on Sobolev Spaces and the Calderon–Vaillancourt Theorem
6.6. Hypercontractivity and Logarithmic Sobolev Inequalities
6.7. Lieb–Thirring and Cwikel–Lieb–Rosenblum Inequalities
6.8. Restriction to Submanifolds
6.9. Tauberian Theorems
Bibliography
Symbol Index
Subject Index
Author Index
Index of Capsule Biographies
