The semiadditivity of continuous analytic capacity and the inner boundary conjecture
Copyright policy )The semiadditivity of continuous analytic capacity and the inner boundary conjecture
Let α( E ) be the continuous analytic capacity of a compact set E ⊂ [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="01i" /]. In this paper we obtain a characterization of α in terms of curvature of measures with zero linear density, and we deduce that α is countably semiadditive. This result has important consequences for the theory of uniform rational approximation on compact sets. In particular, it implies the so-called inner boundary conjecture.
bounded analytic function, Menger curvature, Singular and oscillatory integrals (Calderón-Zygmund, etc.), Capacity and harmonic measure in the complex plane, continuous analytic capacity, Blaschke products, etc., Approximation in the complex plane, rational approximation, Cauchy transform, inner boundary conjecture
bounded analytic function, Menger curvature, Singular and oscillatory integrals (Calderón-Zygmund, etc.), Capacity and harmonic measure in the complex plane, continuous analytic capacity, Blaschke products, etc., Approximation in the complex plane, rational approximation, Cauchy transform, inner boundary conjecture
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