Probabilistic Behavior of Harmonic Functions

1 Basic Ideas and Tools.- 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 R+n+1.- 1.4 The Hardy-Littlewood and nontangential maximal functions.- 1.5 HP spaces on 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.- 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.- 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 the Littlewood-Paley g*-function.- 4 Sharp Good-? Inequalities for A and N.- 4.1 Sharp control of N by A.- 4.2 Sharp control of A by N.- 4.3 Application I. A Chung-type LIL for harmonic functions.- 4.4 Application II. The Burkholder-Gundy ?-theorem.- 5 Good-? Inequalities for the Density of the Area Integral.- 5.1 Sharp control of A and N by D.- 5.2 Sharp control of D by A and N.- 5.3 Application I. A Kesten-type LIL and sharp LP-constants.- 5.4 Application II. The Brossard-Chevalier L log L result.- 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 the Bloch space.- 6.4 On a question of Makarov and Przytycki.- References.- Notation Index.