A characterization of Poissonian domains
type: Item
instance of: scholarly article
title: A characterization of Poissonian domains
zbMATH Open document ID: 0733.31005
DOI: 10.1007/BF02384328
author: Christopher J. Bishop
published in: Arkiv för Matematik
publication date: 1991-01-01T00:00:00Z
review text: A domain \(\Omega \subset {\mathbb{R}}^ n\) is called Poissonian if every bounded harmonic function on \(\Omega\) is the harmonic extension of some function in \(L^{\infty}\) of harmonic measure. It is shown that \(\Omega\) is Poissonian if and only if for every pair of disjoint subdomains \(\Omega_ 1\) and \(\Omega_ 2\) of \(\Omega\) with \(\partial \Omega_ 1\cap \partial \Omega_ 2\subset \partial \Omega\), the harmonic measures of \(\Omega_ 1\) and \(\Omega_ 2\) are mutually singular. Some corollaries of that assertion are proved. It is shown, for example, that each component of the intersection of two Poissonian domains is Poissonian and that if \(E\subset {\mathbb{R}}^ n\) is a closed subset of a Lipschitz graph, then \(\Omega ={\mathbb{R}}^ n\setminus E\) is Poissonian iff E has zero n-1 dimensional measure.
Mathematics Subject Classification ID: 31B10
zbMATH DE Number: 4212421
zbMATH Keywords: bounded harmonic function
zbMATH Keywords: harmonic measures
zbMATH Keywords: Poissonian domains
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4212421