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

María J. González

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arXiv · 2207.04752

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

María J. González

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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.

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María J. González. 2022-07-11. Quasiconformal properties of $Q_{p,0}$ curves and Dirichlet-type curves. https://arxiv.org/abs/2207.04752

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