TAILIEUCHUNG - Đề tài " Bilipschitz maps, analytic capacity, and the Cauchy integral "

Let ϕ : C → C be a bilipschitz map. We prove that if E ⊂ C is compact, and γ(E), α(E) stand for its analytic and continuous analytic capacity respectively, then C −1 γ(E) ≤ γ(ϕ(E)) ≤ Cγ(E) and C −1 α(E) ≤ α(ϕ(E)) ≤ Cα(E), where C depends only on the bilipschitz constant of ϕ. Further, we show that if µ is a Radon measure on C and the Cauchy transform is bounded on L2 (µ), then the Cauchy transform is also bounded on L2 (ϕ µ), where ϕ µ is the image measure of µ by. | Annals of Mathematics Bilipschitz maps analytic capacity and the Cauchy integral By Xavier Tolsa Annals of Mathematics 162 2005 1243-1304 Bilipschitz maps analytic capacity and the Cauchy integral By Xavier Tolsa Abstract Let ip C C be a bilipschitz map. We prove that if E c C is compact and y E a E stand for its analytic and continuous analytic capacity respectively then C-1Y E Y p E Cy E and C-1a E a p E Ca E where C depends only on the bilipschitz constant of . Further we show that if 1 is a Radon measure on C and the Cauchy transform is bounded on L2 i then the Cauchy transform is also bounded on L2 h where Pịp is the image measure of 1 by . To obtain these results we estimate the curvature of Pịp by means of a corona type decomposition. 1. Introduction A compact set E c C is said to be removable for bounded analytic functions if for any open set Q containing E every bounded function analytic on Q E has an analytic extension to Q. In order to study removability in the 1940 s Ahlfors Ah introduced the notion of analytic capacity. The analytic capacity of a compact set E c C is y e sup to where the supremum is taken over all analytic functions f C E C with If I 1 on C E and f TO lim. x z f z - f to . In Ah Ahlfors proved that E is removable for bounded analytic functions if and only if Y E 0. Painleve s problem consists of characterizing removable singularities for bounded analytic functions in a metric geometric way. By Ahlfors result this is equivalent to describing compact sets with positive analytic capacity in metric geometric terms. Partially supported by the program Ramon y Cajal Spain and by grants BFM2000-0361 and MTM2004-00519 Spain 2001-SGR-00431 Generalitat de Catalunya and HPRN-2000-0116 European Union . 1244 XAVIER TOLSA Vitushkin in the 1950 s and 1960 s showed that analytic capacity plays a central role in problems of uniform rational approximation on compact sets of the complex plane. Further he introduced the continuous analytic capacity a .

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