Bilipschitz maps, analytic capacity, and the Cauchy integral

Abstract

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}γ(E)\leq γ(\vphi(E)) \leq Cγ(E) and C^{-1}α(E)\leq α(\vphi(E)) \leq Cα(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(μ), then the Cauchy transform is also bounded on L^2(\vphi_\sharpμ), where vphi_\sharpμis the image measure of mu by vphi. To obtain these results, we estimate the curvature of vphi_\sharpμby means of a corona type decomposition.