Bilipschitz maps, analytic capacity, and the Cauchy integral | Annals of Mathematics

Abstract 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 φ. 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.

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数学年刊(Annals of Mathematics)

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