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

Bilipschitz maps, analytic capacity, and the Cauchy integral