Bilipschitz maps, analytic capacity, and the Cauchy integral

• 成果类型: Article
• 署名作者: Tolsa, X
• 刊物名称: ANNALS OF MATHEMATICS
• ISSN/ISSBN: 0003-486X
• DOI: 10.4007/annals.2005.162.1243
• 发表日期: 2005
• 页码: 1243-1304
• 关键词: semiadditivity; CURVATURE; operator

摘要

Let phi : C -> C be a bilipschitz map. We prove that if E c 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(,U), 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.