On the analytic and Cauchy capacities (Q1661297)

Let \(E\) be a compact subset of \(\mathbb C\). The analytic capacity of \(E\) is defined by \[ \gamma(E)=\sup\{|f'(\infty)|:f\in H^{\infty}(\hat{\mathbb C}\setminus E),\, \|f\|_{\infty}\leq 1 \} \] and the Cauchy capacity of \(E\) is defined by \[ \gamma_{c}(E)=\{|\mu(E)|:\operatorname{supp}(\mu)\subset E \text{ and }|C_{\mu}|\leq 1 \text{ on }\hat{\mathbb C}\setminus E\}, \] where \[ C_{\mu}(z)=\int\frac{1}{\zeta-z}d\mu(\zeta) \] is the Cauchy transform of a complex measure \(\mu\). The author studies the following question: does analytic capacity equal Cauchy capacity? It is well known that the answer is positive for compact sets with finite Painlevé length. In his main result the author proves that this is also the case for sets of \(\sigma\)-finite Painlevé length. Theorem. Let \(E\) be a compact subset of \(\mathbb C\) and suppose that there exists a sequence \(\{E_{k}\}\) of compact subsets of \(E\) such that {\parindent=8mm \begin{itemize}\item[(i)] every \(E_{k}\) has finite Painlevé length, \item[(ii)] there exists an integer \(m\) such that \(\Omega\) and every \(\Omega_{k}\) are nondegenerate \(m\)-connected domains, where \(\Omega\) and \(\Omega_{k}\) are the unbounded components of \(\hat{\mathbb C}\setminus E\) and \(\hat{\mathbb C}\setminus E_{k}\), respectively, \item[(iii)] \(\Omega_{k}\) converges to \(\Omega\) in the sense of Carathéodory. \end{itemize}} Then \(\gamma(E)=\gamma_{c}(E)\). Also, using the above theorem, the author proves the following result about the Ahlfors function. Theorem. For any integer \(m\) there exists a compact set \(E^{m}\) with \(m\)-nondegenerate components such that \(\gamma(E^{m})=\gamma_{c}(E^{m})\) but the Ahlfors function for \(E^{m}\) is not the Cauchy transform of any complex measure supported on \(E^{m}\).