Bilipschitz maps, analytic capacity, and the Cauchy integral

Xavier Tolsa
2005 Annals of Mathematics  
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 where C depends only on the bilipschitz constant of ϕ. Further, we show that if µ is a Radon measure on C and the Cauchy transform is bounded on L 2 (µ), then the Cauchy transform is also bounded on L 2 (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
more » ... of a corona type decomposition.
doi:10.4007/annals.2005.162.1243 fatcat:pa3lhokld5ezrlb5wdfr2dlbci