Potential theory - selected topics
type: Item
instance of: scholarly article
title: Potential theory - selected topics
zbMATH Open document ID: 0865.31001
author: Hiroaki Aikawa
author: Matts Essén
published in: Lecture Notes in Mathematics
publication date: 1996-11-14T00:00:00Z
review text: This book is in two parts, separately authored and indexed, and with separate bibliographies. Part II, by \textit{H. Aikawa}, is a unified account of some recent developments in potential theory due mainly to the author. Part I, by \textit{M. Essén}, presents the necessary potential theoretic background in an informal style, with questions to think about, side remarks and helpful diagrams. The reader is assumed to be familiar with integration and distribution theory, and some functional analysis. The approach of Part I is similar in spirit to the texts of \textit{L. Carleson} [Selected problems on exceptional sets, Van Nostrand, Princeton (1967; Zbl 0189.10903); reprint Wadsworth (1983; Zbl 0505.00034)] and \textit{N. S. Landkof} [Foundations of modern potential theory, Springer, Berlin (1972; Zbl 0253.31001)]. It does not attempt to be a comprehensive treatise on potential theory, but does cover, in some form, a substantial part of the core of the subject. Where proofs are omitted, a convenient reference is supplied. The account begins with a discussion of potentials and capacity arising from a certain class of kernels. The text presents several equivalent formulations of capacity, the capacitability of analytic sets, the relationship between Hausdorff measures and capacities, and Cantor sets which illustrate the subtleties of this relationship. Attention is then directed to potentials arising from the M. Riesz kernel \(k_\alpha(x)= |x|^{\alpha-N}\), where \(0< \alpha < N\). There is an account of \(\alpha\)-energy, \(\alpha\)-capacity and associated equilibrium measures. For the potential theory of the Laplace operator, there is a discussion of Green potentials and reduced functions, the fine topology and thin sets. Then, in the context of a ball or half-space, minimally thin and rarefied sets are introduced and characterized by Wiener-type criteria involving Green energy and Green mass, respectively. Singular integral methods are used to establish the Beurling-Ma…
reviewed by: Stephen J. Gardiner
Mathematics Subject Classification ID: 31-02
Mathematics Subject Classification ID: 31B15
Mathematics Subject Classification ID: 31B05
Mathematics Subject Classification ID: 31A05
Mathematics Subject Classification ID: 31B25
zbMATH DE Number: 945655
zbMATH Keywords: \(\alpha\)-energy
zbMATH Keywords: \(\alpha\)-capacity
zbMATH Keywords: singular integral methods
zbMATH Keywords: \(L^ p\)-potential theory
zbMATH Keywords: rarefied sets
zbMATH Keywords: capacity
zbMATH Keywords: Hausdorff measures
zbMATH Keywords: Cantor sets
zbMATH Keywords: Riesz kernel
zbMATH Keywords: potential theory of the Laplace operator
zbMATH Keywords: Green potentials
zbMATH Keywords: reduced functions
zbMATH Keywords: fine topology
zbMATH Keywords: thin sets
zbMATH Keywords: Wiener-type criteria
zbMATH Keywords: Green mass
zbMATH Keywords: Beurling-Mazya-Dahlberg result
zbMATH Keywords: Green energy
zbMATH Keywords: Whitney-type decompositions
zbMATH Keywords: capacity strong-type inequality
zbMATH Keywords: Riesz capacity
zbMATH Keywords: boundary behaviour of harmonic and superharmonic functions
zbMATH Keywords: Fatou-Naïm-Doob minimal fine limit theorem
zbMATH Keywords: integrability of superharmonic and subharmonic functions
MaRDI profile type: Publication
DOI: 10.1007/BFB0093410
description: scientific article; zbMATH DE number 945655