Bilipschitz maps, analytic capacity, and the Cauchy integral

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