On the analytic and Cauchy capacities

TL;DR

This paper establishes new conditions under which the analytic and Cauchy capacities of a compact set are equal, and provides examples where this equality holds without the Ahlfors function being a Cauchy transform of a measure.

Contribution

It introduces novel sufficient conditions for the equality of analytic and Cauchy capacities and constructs examples where the Ahlfors function is not a Cauchy transform.

Findings

New sufficient conditions for capacity equality

Examples where the Ahlfors function is not a Cauchy transform

Equality of capacities without measure representation

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.