Overview

Harmonic analysis and probability have enjoyed a mutually beneficial relationship. This monograph explores several aspects of this relationship. The primary focus of the text is the nonangential maximal function and the area function of a harmonic function and their probablistic analogues in martingale theory.

Full Product Details

Author:   Rodrigo Banuelos ,  Charles N. Moore ,  Hyman Bass ,  Joseph Oesterle
Publisher:   Birkhauser Verlag AG
Imprint:   Birkhauser Verlag AG
Edition:   1999 ed.
Volume:   175
Dimensions:   Width: 15.50cm , Height: 1.40cm , Length: 23.50cm
Weight:   1.100kg
ISBN:  

9783764360627

ISBN 10:   3764360623
Pages:   209
Publication Date:   01 August 1999
Audience:   College/higher education ,  Professional and scholarly ,  Postgraduate, Research & Scholarly ,  Professional & Vocational
Format:   Hardback
Publisher's Status:   Active
Availability:   In Print  
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Table of Contents

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.

Reviews

The book is devoted to the interplay of potential theory and probability theorya ]The reader interested in this subject a the interplay of probability theory, harmonic analysis and potential theory a will find a systematic treatment, inspiring both sides, analysis and probability theory. <p>a Zentralblatt Math

The book is devoted to the interplay of potential theory and probability theory...The reader interested in this subject - the interplay of probability theory, harmonic analysis and potential theory - will find a systematic treatment, inspiring both sides, analysis and probability theory. -Zentralblatt Math

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