Computational Methods and Function Theory 7 (2007), No. 1, 91--104
Copyright Heldermann Verlag 2007
John R. Akeroyd
jakeroyd@uark.edu , University of Arkansas, Department of Mathematics, Science-Engineering 301, Fayetteville, AR 72701, U.S.A.
Let $\Omega$ be a subregion of $\{z:\, |z| < 1\}$ for which the Dirichlet problem is solvable, assume that $0\in\Omega$ and let $\omega_\Omega$ denote harmonic measure on $\partial{\Omega}$ for evaluation at $0$. If $E$ is a Borel subset of $\{z:\, |z| = 1\}$ and $\omega_\Omega(E) > 0$, then we find a simply connected region $G$, where $0\in G \subseteq\{z:\, |z| < 1\}$, $\partial{G}\subseteq\Omega\cup E$ and $\omega_G(E)>0$, such that $U := G\cup\Omega$ has the property that $\omega_U$ and $\omega_\Omega$ are boundedly equivalent on $\partial{U}$. We mention consequences of this in function theory.
Keywords: Harmonic measure, Green's potential.
MSC 2000: 31A15, 31A05, 46E15.
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