$L^2$ boundedness of the Cauchy transform implies $L^2$ boundedness of all antisymmetric Calderón-Zygmund operators associated to odd kernels

Abstract

Let µ be a Radon measure on C without atoms. In this paper we prove that if the Cauchy transform is bounded in L2 (µ), then all 1-dimensional Calder´on-Zygmund operators associated to odd and sufficiently smooth kernels are also bounded in L2 (µ).