A characterization of Poissonian domains (Q810198)

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.