Endpoint compactness of singular integrals and perturbations of the Cauchy Integral

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