Quasiconformal properties of Q(p),0 curves and Dirichlet-type curves

Abstract

Let Gamma be a closed Jordan curve, and f the conformal mapping that sends the unit disc D onto the interior domain of Gamma. If log f ' belongs to the Dirichlet space 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 Qp,0, space, for 0 < p <= 1, or to some weighted-Dirichlet space contained in 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. (c) 2022 The Author(s). Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).