On the analytic and Cauchy capacities

Abstract

We give new sufficient conditions for a compact set $E\subseteq \mathbb{C}$ to satisfy $\gamma (E)={\gamma }_c(E)$, where $\gamma$ is the analytic capacity and ${\gamma }_c$ is the Cauchy capacity. As a consequence, we provide examples of compact plane sets such that the above equality holds but the Ahlfors function is not the Cauchy transform of any complex Borel measure supported on the set.

16 pages, 6 figures