Quasiconformal properties of $Q_{p,0}$ curves and Dirichlet-type curves

TL;DR

This paper extends the understanding of geometric properties and quasiconformal extensions of curves associated with conformal maps whose derivatives' logarithm belongs to certain function spaces, generalizing previous Weil-Petersson curve results.

Contribution

It generalizes recent results on Weil-Petersson curves to cases where the logarithm of the conformal map's derivative is in $Q_{p,0}$ or weighted-Dirichlet spaces, characterizing quasiconformal extensions.

Findings

Characterization of quasiconformal extensions for these curves

Description of geometric properties of the curves

Extension of results from Weil-Petersson to broader function spaces

Abstract

Let $\Gamma$ be a closed Jordan curve, and $f$ the conformal mapping that sends the unit disc $\mathbb{D}$ onto the interior domain of $\Gamma$. If $\log f'$ belongs to the Dirichlet space $\mathcal{D}$, we call $\Gamma$ a Weil-Petersson curve. The purpose of this note is to extend recent results, obtained by G. Cui and Ch. Bishop in the case of Weil-Petersson curves, to the case when $\log f'$ belongs to either some $Q_{p,0},$ space, for $0<p\leq 1$, or to some weighted-Dirichlet space contained in $\mathcal{D}$. More precisely, we will characterize the quasiconformal extensions of $f$, and describe some of the geometric properties of $\Gamma$, that arise in this context.