一个费米子 Grunsky 算子
A Fermionic Grunsky operator
AI总结：
本文针对边界正则的共形映射引入了费米子 Grunsky 算子，证明其拉回的 Smirnov 空间为算子图像，并刻画了其满足 Hilbert-Schmidt 条件时与 Weil-Petersson 拟圆盘的等价关系。
摘要：
对于从圆盘 $\mathbb{D}$ 映入复平面且像域具有可求长 Ahlfors-正则边界的共形映射 $f$，我们在单位圆盘的 Hardy 空间上关联了一种新的 Grunsky 算子。这类似于经典的 Grunsky 算子，后者本身可视为 Bergman 或 Dirichlet 空间上的算子。我们证明了 $f(\mathbb{D})$ 补集的 Smirnov 空间被 $f$ 拉回后即为该 Grunsky 算子的图像。我们还刻画了具有可求长 Ahlfors-正则边界且使 Grunsky 算子为 Hilbert-Schmidt 算子的区域。特别地，我们证明了若 Grunsky 算子是 Hilbert-Schmidt 的，则 $f(\mathbb{D})$ 是一个 Weil-Petersson 拟圆盘。这些结果的表述与证明本质上利用了对 Smirnov 空间作为半阶微分空间的几何处理。
英文摘要：
To a conformal map $f$ from the disk $\mathbb{D}$ into the complex plane onto a domain with rectifiable Ahlfors-regular boundary, we associate a new kind of Grunsky operator on the Hardy space of the unit disk. This is analogous to the classical Grunsky operator, which itself can be viewed as an operator on Bergman or Dirichlet space. We show that the pull-back of the Smirnov space of the complement of $f(\mathbb{D})$ by $f$ is the graph of the Grunsky operator. We also characterize those domains with rectifiable Ahlfors-regular boundaries such that the Grunsky operator is Hilbert-Schmidt. In particular, we show that if the Grunsky operator is Hilbert-Schmidt, then $f(\mathbb{D})$ is a Weil-Petersson quasidisk. The formulations of the results and proofs make essential use of a geometric treatment of Smirnov space as a space of half-order differentials.
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