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.


30C85 Capacity and harmonic measure in the complex plane
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