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 Hp-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 Lp-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 Lp-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