Painleve's problem and the semiadditivity of analytic capacity

Abstract

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.

42 pages