Cauchy independent measures and super-additivity of analytic capacity

TL;DR

This paper investigates the super-additivity of analytic capacity for subsets of discs centered on a smooth curve, establishing conditions under which capacities add up and exploring implications for Cauchy integral operators.

Contribution

It introduces a new super-additivity result for analytic capacity under a separation condition and applies it to analyze the independence of Cauchy integral operators.

Findings

Analytic capacities of subsets of separated discs add up.

Separation condition is crucial for super-additivity.

Results impact understanding of Cauchy integral operator independence.

Abstract

We show that, given a family of discs centered at a nice curve, the analytic capacities of arbitrary subsets of these discs add up. However we need that the discs in question would be slightly separated, and it is not clear whether the separation condition is essential or not. We apply this result to study the independence of Cauchy integral operators.