Ahlfors-regular curves and Carleson measures

TL;DR

This paper explores the connection between Ahlfors-regular boundary curves and Carleson measures, providing characterizations of certain geometric curves via quasiconformal mappings and complex dilatation.

Contribution

It establishes a link between boundary regularity and Carleson measure invariance, and characterizes specific curves using quasiconformal reflection properties.

Findings

Ahlfors-regular boundary implies invariance of Carleson measures under conformal maps.

Chord-arc curves with small norm are characterized via complex dilatation.

Asymptotically smooth curves are described through quasiconformal reflection properties.

Abstract

We study the relation between the boundary of a simply connected domain being Ahlfors-regular and the invariance of Carleson measures under the push-forward operator induced by a conformal mapping from the unit disk onto the domain. As an application, we characterize the chord-arc curve with small norm and the asymptotically smooth curve in terms of the complex dilatation of some quasiconformal reflection with respect to the curve.