Toshiyuki SUGAWA

### Abstract

In this paper, we propose a notion of the local harmonic measure decay (LHMD) property with exponent $\alpha$ at finite boundary points of open sets $\Omega$ in the Riemann sphere $\widehat{\mathbb{C}}.$ Using this property, we show that Green's function of $\Omega$ is H\"older continuous with exponent $\alpha$ at such a point as well as the boundary regularity of the Dirichlet problem in $\Omega$ for the usual Laplacian at the point in the sense of H\"older continuity with exponent less than $\alpha.$ We further explain that the LHMD property can be regarded as a localization of the notion of uniform perfectness for the boundary. We also provide several applications to the theory of conformal mappings.

submission: April 15, 1999
revision: April 16, 1999
revision: June 1, 1999
revision: May 30, 2001

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