Cauchy transforms and Szeg\H{o} projections in dual Hardy spaces: inequalities and M\"obius invariance

TL;DR

This paper investigates the properties of Cauchy transforms and Szeg\

Contribution

It introduces a M"obius invariant function bounding the Cauchy transform norm and explores its rigidity, connecting it to the Kerzman-Stein operator with explicit examples.

Findings

A M"obius invariant norm bound for the Cauchy transform

Rigidity properties of the invariant function

New asymptotically sharp lower bound for ellipses

Abstract

Dual pairs of interior and exterior Hardy spaces associated to a simple closed Lipschitz planar curve are considered, leading to a M\"obius invariant function bounding the norm of the Cauchy transform $\bf{C}$ from below. This function is shown to satisfy strong rigidity properties and is closely connected via the Berezin transform to the square of the Kerzman-Stein operator. Explicit example calculations are presented. For ellipses, a new asymptotically sharp lower bound on the norm of $\bf{C}$ is produced.