Abstract. A compact set E ⊂ C is said to be removable for bounded analytic functions if for any open set Ω containing E, every bounded function analytic on Ω\E has an analytic extension to Ω. Analytic capacity is a notion that, in a sense, measures the size of a set as a non removable singularity. In particular, a compact set is removable if and only if its analytic capacity vanishes. The so called Painlevé problem consists in characterizing removable sets in geometric terms. Recently many results in connection with this very old and challenging problem have been obtained. Moreover, it has also been proved that analytic capacity is semiadditive. We review these results and other related questions dealing with rectifiability, the Cauchy transform, and the Riesz transforms.
Mathematics Subject Classification (2000). Primary 30C85; Secondary 42B20, 28A75.
The analytic capacity γ + is a version of the usual analytic capacity γ which is generated by Cauchy potentials of positive measures. Some recent results have shown the importance of γ + for the understanding of the metric-geometric properties of γ. This paper is devoted to the study of γ +. Among other things, it is shown that although this capacity is not originated by a positive symmetric kernel, it satisfies some properties usually fulfilled this other type of capacities (such as Riesz capacities). 10 9 c 2 ν (y) + C 3 Mν(y) 2 , and c 2 ν (x) ≤ 10 9 c 2 ν (y) + C 3 Mν(x) 2. The proof of Lemma (3.5) follows by standard arguments. To prove Theorem 3.3 we will need to apply a variational argument on some 'nice' approximation of E by another compactẼ. The following lemma will be very useful.
Let µ be a Radon measure on C without atoms. In this paper we prove that if the Cauchy transform is bounded in L 2 (µ), then all 1-dimensional Calderón-Zygmund operators associated to odd and sufficiently smooth kernels are also bounded in L 2 (µ).2000 Mathematics Subject Classification. 42B20.
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