Painleve's problem and the semiadditivity of analytic capacity

Details Painleve's problem and the semiadditivity of analytic capacity Painleve's problem and the semiadditivity of analytic capacity

2002-04-02

arxiv.org/abs/math/0204027v1

Mathematics

Classical Analysis and ODEs

42 pages

Scientific paper

Let $\gamma(E)$ be the analytic capacity of a compact set $E$ and let $\gamma_+(E)$ be the capacity of $E$ originated by Cauchy transforms of positive measures. In this paper we prove that $\gamma(E)\approx\gamma_+(E)$ with estimates independent of $E$. As a corollary, we characterize removable singularities for bounded analytic functions in terms of curvature of measures, and we deduce that $\gamma$ is semiadditive, which solves a long standing question of Vitushkin.

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