Bilipschitz maps, analytic capacity, and the Cauchy integral

Abstract

Let $\varphi:\mathbb{C}\rightarrow \mathbb{C}$ be a bilipschitz map. We prove that if $E\subset\mathbb{C}$ is compact, and $\gamma(E)$, $\alpha(E)$ stand for its analytic and continuous analytic capacity respectively, then $C^{-1}\gamma(E)\leq \gamma(\varphi(E)) \leq C\gamma(E)$ and $C^{-1}\alpha(E)\leq \alpha(\varphi(E)) \leq C\alpha(E)$, where $C$ depends only on the bilipschitz constant of $\varphi$. Further, we show that if $\mu$ is a Radon measure on $\mathbb{C}$ and the Cauchy transform is bounded on $L^2(\mu)$, then the Cauchy transform is also bounded on $L^2(\varphi_\sharp\mu)$, where $\varphi_\sharp\mu$ is the image measure of $\mu$ by $\varphi$. To obtain these results, we estimate the curvature of $\varphi_\sharp\mu$ by means of a corona type decomposition.