Bilipschitz maps, analytic capacity, and the Cauchy integral

Details Bilipschitz maps, analytic capacity, and the Cauchy integral Bilipschitz maps, analytic capacity, and the Cauchy integral

2003-03-19

arxiv.org/abs/math/0303243v2

Ann. of Math. (2) 162 (2005), no. 3, 1243--1304

Mathematics

Classical Analysis and ODEs

62 pages, published version

Scientific paper

Let vphi:C rightarrow C be a bilipschitz map. We prove that if E\subset\C is compact, and gamma(E), alpha(E) stand for its analytic and continuous analytic capacity respectively, then C^{-1}\gamma(E)\leq \gamma(\vphi(E)) \leq C\gamma(E) and C^{-1}\alpha(E)\leq \alpha(\vphi(E)) \leq C\alpha(E), where C depends only on the bilipschitz constant of vphi. Further, we show that if mu is a Radon measure on C and the Cauchy transform is bounded on L^2(\mu), then the Cauchy transform is also bounded on L^2(\vphi_\sharp\mu), where vphi_\sharp\mu is the image measure of mu by vphi. To obtain these results, we estimate the curvature of vphi_\sharp\mu by means of a corona type decomposition.

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