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A Note on Harmonic Measure

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Abstract

Let Ω be a subregion of {z : |z| < 1} for which the Dirichlet problem is solvable, assume that 0 ∈ Ω and let ω Ω denote harmonic measure on ∂Ω for evaluation at 0. If E is a Borel subset of {z : |z| = 1} and ω Ω (E) > 0, then we find a simply connected region G, where 0 ∈ G ⊆ {z : |z| < 1}, ∂G ⊆ Ω ∪ E and ω G (E) > 0, such that U := G ∪ Ω has the property that w U and w Ω are boundedly equivalent on ∂U. We mention consequences of this in function theory.

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  1. Department of Mathematics, University of Arkansas, Science-Engineering 301, Fayetteville, AR 72701, USA

    John R. Akeroyd

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  1. John R. Akeroyd
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Correspondence to John R. Akeroyd.

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Akeroyd, J.R. A Note on Harmonic Measure. Comput. Methods Funct. Theory 7, 91–104 (2007). https://doi.org/10.1007/BF03321633

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