Measures that define a compact Cauchy transform

TL;DR

This paper characterizes measures with compact support on the plane for which the associated Cauchy transform acts as a compact operator on L^2 spaces, emphasizing the roles of measure density and Menger curvature.

Contribution

It provides a geometric characterization of measures ensuring the Cauchy transform is a compact operator, highlighting the importance of measure density and Menger curvature.

Findings

Measures with certain density conditions yield compact Cauchy transforms.

Menger curvature is a key factor in the compactness characterization.

The work links geometric measure properties to operator compactness.

Abstract

The aim of this work is to provide a geometric characterization of the positive Radon measures $\mu$ with compact support on the plane such that the associated Cauchy transform defines a compact operator from $L^2(\mu)$ to $L^2(\mu).$ It turns out that a crucial role is played by the density of the measure and by its Menger curvature.