Painlevé's problem and the semiadditivity of analytic capacity
Copyright policy )Painlevé's problem and the semiadditivity of analytic capacity
Let $��(E)$ be the analytic capacity of a compact set $E$ and let $��_+(E)$ be the capacity of $E$ originated by Cauchy transforms of positive measures. In this paper we prove that $��(E)\approx��_+(E)$ 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 $��$ is semiadditive, which solves a long standing question of Vitushkin.
42 pages
Singular and oscillatory integrals (Calderón-Zygmund, etc.), \(T(b)\)-theorem, 30C85; 42B20, Capacity and harmonic measure in the complex plane, Potentials and capacity, harmonic measure, extremal length and related notions in two dimensions, 30C85, Cauchy transform, analytic capacity \(\gamma_+\), Mathematics - Classical Analysis and ODEs, bounded holomorphic function, analytic capacity, Classical Analysis and ODEs (math.CA), FOS: Mathematics, 42B20, Function spaces arising in harmonic analysis, curvature of measures
Singular and oscillatory integrals (Calderón-Zygmund, etc.), \(T(b)\)-theorem, 30C85; 42B20, Capacity and harmonic measure in the complex plane, Potentials and capacity, harmonic measure, extremal length and related notions in two dimensions, 30C85, Cauchy transform, analytic capacity \(\gamma_+\), Mathematics - Classical Analysis and ODEs, bounded holomorphic function, analytic capacity, Classical Analysis and ODEs (math.CA), FOS: Mathematics, 42B20, Function spaces arising in harmonic analysis, curvature of measures
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