Endpoint compactness of singular integrals and perturbations of the Cauchy Integral

Abstract

We prove sufficient and necessary conditions for compactness of Calderón-Zygmund operators on the endpoint from $L^{\infty }(\mathbb{R})$ into $\mathrm{CMO}(\mathbb{R})$. 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 $\mathrm{CMO}$-condition.

arXiv admin note: text overlap with arXiv:1211.0672