Endpoint compactness of singular integrals and perturbations of the Cauchy integral

Abstract

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