PROBABILISTIC BEHAVIOR OF HARMONIC FUNCTIONS
(Birkhäuser 1999)
Rodrigo Bañuelos and Charls N. Moore

TABLE OF CONTENTS

• PREFACE
• CHAPTER 1. Introduction
  • §1.1 Harmonic Functions and their basic properties
  • §1.2 The Poisson kernel and Dirichlet problem for the ball
  • §1.3 The Poisson kernel and Dirichlet problem for the upper-half space
  • §1.4 The Hardy-Littlewood and nontangential maximal functions
  • §1.5 H^p-theory for the upper half-space
  • §1.6 Some basics on singular integrals
  • §1.7 The g-function and area function
  • §1.8 Classical results on boundary behavior
• CHAPTER 2. Decomposition into Martingales: An Invariance Principle
  • §2.1. Square function estimates for sums of atoms
  • §2.2. Decomposition of harmonic functions
  • §2.3. Controlling errors: gradient estimates
• CHAPTER 3. Kolmogorov's LIL for Harmonic Functions
  • §3.1. The proof of the upper-half
  • §3.2. The proof of the lower-half
  • §3.3. The sharpness of the Kolmogorov condition
  • §3.4. A related LIL for Littlewood-Paley square functions
• CHAPTER 4. Sharp good-lambda Inequalities for A and N
  • §4.1. Sharp control of N by A
  • §4.2. Sharp control of A by N
  • §4.3. Applications I: A Chung-type LIL for harmonic functions
  • §4.4. Applications II: Sharp L^p-constants and ratio inequalities
• CHAPTER 5. Sharp good-lambda Inequalities for the Density of the Area Integral
  • §5.1. Sharp control of N and A by D
  • §5.2. Sharp control of D by N and A
  • §5.3. Applications I: A Kesten-Type LIL and sharp L^p-constants
  • 5.4. Applications II: The Brossard-Chevalier LlogL result
• CHAPTER 6. The classical LIL's in Analysis
  • §6.1. LIL's for lacunary series
  • §6.2. LIL's for Bloch function
  • §6.3. LIL's for subclasses of Bloch functions
  • §6.4. On a question of Makarov and Przytycki
• REFERENCES
• SUBJECT INDEX
• NOTATION INDEX