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We prove sufficient and necessary conditions for compactness ofCalder\'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 Cauchyintegral on curves with normal derivatives satisfying a ${\rm CMO}$-condition.

Endpoint compactness of singular integrals and perturbations of the Cauchy Integral