Chord-arc curves and the Beurling transform

摘要

We study the relation between the geometric properties of a quasicircle and the complex dilatation of a quasiconformal mapping that maps the real line onto . Denoting by S the Beurling transform, we characterize Bishop-Jones quasicircles in terms of the boundedness of the operator on a particular weighted space, and chord-arc curves in terms of its invertibility. As an application we recover the boundedness of the Cauchy integral on chord-arc curves.