Bilipschitz maps, analytic capacity, and the Cauchy integral

Let (phi)C (right arrow)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)<=(gamma)((phi)(E))<=C(gamma)(E) and C^-1(alpha)(E)<=(alpha)((phi)(E)) <=C(alpha)(E), where C depends only on the bilipschitz constant of (phi). 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((phi)#(mu)), where (phi)#(mu) is the image measure of (mu) by (phi). To obtain these results, we estimate the curvature of (phi)#(mu) by means of a corona type decomposition.