Chord-arc curves and the Beurling transform

TL;DR

This paper explores the relationship between geometric properties of quasicircles and the Beurling transform, providing characterizations of Bishop-Jones quasicircles and chord-arc curves through operator boundedness and invertibility.

Contribution

It introduces new characterizations of quasicircle types via the boundedness and invertibility of a specific operator involving the Beurling transform and complex dilatation.

Findings

Characterization of Bishop-Jones quasicircles through operator boundedness.

Chord-arc curves characterized by invertibility of the operator.

Recovery of L^2 boundedness of the Cauchy integral on chord-arc curves.

Abstract

We study the relation between the geometric properties of a quasicircle~$\Gamma$ and the complex dilatation~$\mu$ of a quasiconformal mapping that maps the real line onto~$\Gamma$. Denoting by~$S$ the Beurling transform, we characterize Bishop-Jones quasicircles in terms of the boundedness of the operator~$(I-\mu S)$ on a particular weighted $L^2$~space, and chord-arc curves in terms of its invertibility. As an application we recover the~$L^2$ boundedness of the Cauchy integral on chord-arc curves.