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
functions
- §6.3. LIL's for
subclasses of Bloch functions
- §6.4. On a question
of Makarov and Przytycki
- REFERENCES
- SUBJECT INDEX
- NOTATION INDEX