Estimates for the Maximal Cauchy Integral on Chord-arc Curves

TL;DR

This paper characterizes chord-arc Jordan curves satisfying a Cotlar-type inequality involving the Cauchy transform and maximal functions, linking geometric properties of the curves to their parametrization smoothness.

Contribution

It provides a new characterization of such curves based on the smoothness of their parametrization under asymptotic quasi-conformality assumptions.

Findings

Characterization of chord-arc curves satisfying Cotlar-type inequality.

Connection between curve smoothness and inequality satisfaction.

Insight into geometric properties influencing Cauchy transform estimates.

Abstract

We study the chord-arc Jordan curves that satisfy the Cotlar-type inequality $T_*(f)\lesssim M^2(Tf),$ where $T$ is the Cauchy transform, $T_*$ is the maximal Cauchy transform and $M$ is the Hardy-Littlewood maximal function. Under the background assumption of asymptotic quasi-conformality we find a characterization of such curves in terms of the smoothness of a parametrization of the curve.