Bilipschitz maps, analytic capacity, and the Cauchy integral.(English)Zbl 1097.30020

A compact set $$E$$ in the complex plane $${\mathbb C}$$ is said to be removable for bounded analytic functions if for any open set $$\Omega\supset E$$, every bounded analytic function in $$\Omega\setminus E$$ has an analytic extension to $$\Omega$$. An important old problem in Complex Analysis, called the Painlevé’s problem, is to find a geometric characterization of removable sets, or, equivalently, to describe compact sets $$E$$ with positive analytic capacity $\gamma(E):=\sup \{| f^\prime(\infty)| :f:{\mathbb C}\setminus E\to {\mathbb C}\;\;\text{analytic,}\;\;\;| f| \leq 1\}.$ The problem has deep connections with rational approximation, the Cauchy transform, and the curvature of measures. Recently, Tolsa and others have obtained important results on Painlevé’s problem; see e.g. X. Tolsa’s survey [”Painlevé’s problem, analytic capacity and curvature of measures”. Proceedings of the 4th European congress of mathematics (ECM), Stockholm, Sweden, June 27-July 2, 2004. European Mathematical Society (EMS). 459-476 (2005; Zbl 1095.30021)].
In the present paper Tolsa proves that removability is invariant under bilipschitz maps. More precisely, he shows that if $$\phi:{\mathbb C}\to {\mathbb C}$$ is a bilipschitz map, then there exists a constant $$C>0$$ such that $C^{-1}\gamma(E)\leq \gamma(\phi(E))\leq C\gamma(E).$ J. Garnett and J. Verdera [Math. Res. Lett. 10, 515-522 (2003; Zbl 1063.30025)] have proved this result for a wide class of Cantor sets $$E$$. Tolsa obtains the above result using his previous work and the following theorem: Let $$\mu$$ be a Radon measure supported on $$E$$ such that $$\mu(D(x,r))\leq r$$ for all $$x\in E$$, $$r>0$$ and $$c^2(\mu)<\infty$$. Let $$\phi$$ be a bilipschitz map. Then for a constant $$C>0$$, $c^2(\phi_\sharp\mu)\leq C(\mu(E)+c^2(\mu))$ Here $$\phi_\sharp\mu$$ is the image measure of $$\mu$$ by $$\phi$$ and $$c^2(\mu)$$ is the curvature of $$\mu$$, a basic tool in the study of Painlevé’s problem, introduced by Melnikov. The proof of this theorem uses a corona type decomposition.
It is also proved that the assumption that $$\phi$$ is bilipschitz is necessary in the sense that if $$\phi$$ is homeomorphism and $$\gamma(E)\approx \gamma(\phi(E))$$, then $$\phi$$ is bilipschitz. Analogous results for the continuous analytic capacity $$\alpha(E)$$ are also proved.

MSC:

 30C85 Capacity and harmonic measure in the complex plane

Citations:

Zbl 1063.30025; Zbl 1095.30021
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