Endpoint compactness of singular integrals and perturbations of the Cauchy Integral

TL;DR

This paper establishes criteria for the compactness of Calderón-Zygmund operators at the endpoint from $L^{inity}$ to CMO, and applies these to perturbations of the Cauchy integral on curves with specific boundary conditions.

Contribution

It provides necessary and sufficient conditions for endpoint compactness of Calderón-Zygmund operators and demonstrates their application to perturbed Cauchy integrals on certain curves.

Findings

Characterization of compactness conditions for Calderón-Zygmund operators.

Proof of compactness for perturbations of the Cauchy integral on curves with CMO boundary derivatives.

Extension of endpoint compactness results to a broader class of singular integral operators.

Abstract

We prove sufficient and necessary conditions for compactness of Calder\'on-Zygmund operators on the endpoint from $L^{\infty }(\mathbb R)$ into ${\rm CMO}(\mathbb R)$. We use this result to prove compactness on $L^{p}(\mathbb R)$ with $1<p<\infty $ of certain perturbations of the Cauchy integral on curves with normal derivatives satisfying a ${\rm CMO}$-condition.