$Q_{p,0}$曲线和Dirichlet型曲线的拟共形性质
Quasiconformal properties of $Q_{p,0}$ curves and Dirichlet-type curves
摘要：
设$Γ$是一条封闭的Jordan曲线，$f$是将单位圆$\mathbb{D}$映射到$Γ$的内部区域上的共形映射。若$\log f'$属于Dirichlet空间$\mathcal{D}$，则称$Γ$为Weil-Petersson曲线.本文的目的是推广G. Cui和Ch. Bishop在Weil-Petersson曲线的情况下，到$\log f'$属于某个$Q_{p,0},$空间（对于$0<p\leq 1$）或属于包含在$\mathcal{D}$中的某个加权Dirichlet空间的情况。更准确地说，我们将描述拟共形扩展的$f$，并描述一些几何性质的$Γ$，出现在这种情况下。
英文摘要：
Let $Γ$ be a closed Jordan curve, and $f$ the conformal mapping that sends the unit disc $\mathbb{D}$ onto the interior domain of $Γ$. If $\log f'$ belongs to the Dirichlet space $\mathcal{D}$, we call $Γ$ 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 $Γ$, that arise in this context.
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