Bilipschitz maps, analytic capacity, and the Cauchy integral

Let . : C . C be a bilipschitz map. We prove that if E . C is compact, and .(E), a(E) stand for its analytic and continuous analytic capacity respectively, then C-1.(E) = .(.(E)) = C.(E) and C-1a(E) = a(.(E)) = Ca(E), where C depends only on the bilipschitz constant of .. 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(.µ), where .µ is the image measure of µ by .. To obtain these results, we estimate the curvature of .µ by means of a corona type decomposition