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Painlevé’s problem and analytic capacity. (English) Zbl 1105.30015
This is a survey of some recent results in connection with the Painlevé’s problem, the semiadditivity of analytic capacity and other related questions. Painlevé’s problem consists in characterizing removable singularities of bounded analytic functions in a geometric/metric way. Equivalently, the problem is to characterize compact sets of zero analytic capacity. Recently, the author proved the semiadditivity of analytic capacity. This property, combined with other recent results, led to important progress in Painlevé’s problem. The present paper contains a detailed review of these recent results. It also includes the proof of the semiadditivity of analytic capacity.

MSC:
30C85 Capacity and harmonic measure in the complex plane
31A15 Potentials and capacity, harmonic measure, extremal length and related notions in two dimensions
42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.)
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