Author | Baรฑuelos, Rodrigo. author |
---|---|
Title | Probabilistic Behavior of Harmonic Functions [electronic resource] / by Rodrigo Baรฑuelos, Charles N. Moore |
Imprint | Basel : Birkhรคuser Basel : Imprint: Birkhรคuser, 1999 |
Connect to | http://dx.doi.org/10.1007/978-3-0348-8728-1 |
Descript | XIV, 209 p. online resource |
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 Kolmogorovโs 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 LILโs in Analysis -- 6.1 LILโs for lacunary series -- 6.2 LILโs for Bloch functions -- 6.3 LILโs for subclasses of the Bloch space -- 6.4 On a question of Makarov and Przytycki -- References -- Notation Index