Wiener's type regularity criteria on the complex plane

Józef Siciak

ArticleOriginal scientific text

Title

Wiener's type regularity criteria on the complex plane

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Affiliations

Institute of Mathematics Jagiellonian University Reymonta 4 30-059 Kraków, Poland

Abstract

We present a number of Wiener's type necessary and sufficient conditions (in terms of divergence of integrals or series involving a condenser capacity) for a compact set E ⊂ ℂ to be regular with respect to the Dirichlet problem. The same capacity is used to give a simple proof of the following known theorem [2, 6]: If E is a compact subset of ℂ such that

d (t^{- 1} E \cap {| z - a | \leq 1}) \geq c o n s t > 0

for 0 < t ≤ 1 and a ∈ E, where d(F) is the logarithmic capacity of F, then the Green function of ℂ \ E with pole at infinity is Hölder continuous.

Keywords

ENG

subharmonic functions, logarithmic potential theory, Green function, regular points, Hölder Continuity Property

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Title

Wiener's type regularity criteria on the complex plane

Affiliations

Abstract

Keywords

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