AI 中文总结
本文研究与单位圆盘压缩算子关联的狄利克雷符号，分析其满足的非线性波动方程，引入Schwarzian渐近方差并得到相关振幅的上界，扩展了通用Teichmüller空间的适用范围。
AI 中文摘要
我们研究Hedenmalm和Shimorin（2020）引入的狄利克雷型算子符号，该符号与单位圆盘上的给定压缩算子相关，且始终是双圆盘上的全纯函数。与圆盘或外圆盘上单叶函数的Grunsky算子关联的此类狄利克雷符号具有特殊意义，据Hedenmalm和Shimorin的研究，其特征为某一非线性波动方程的解。我们对双圆盘对角附近的此类符号进行局部分析，借此为圆盘或外圆盘上单叶函数的无限维流形提供了替代图表坐标，这些坐标可在不直接涉及单叶性的情况下刻画Σ类（归一化单叶函数类）中ψ的logψ'。此外，这些流形将Lipman Bers的通用Teichmüller空间扩展到拟圆周边界设定之外，允许存在更多分形结构。可通过McMullen（2008）引入的渐近方差研究与给定单叶函数关联的区域的调和测度的分形性，该渐近方差捕获非线性性的L²平均振幅。我们引入新的Schwarzian渐近方差概念，用以测量Schwarzian导数的平均振幅以替代非线性性；对于该新的Schwarzian渐近方差，我们发现双曲度量意义下圆盘上(1-|z|²)²|S(φ)|²的有效平均振幅至多为72/5=14.4，远小于最大振幅36，其中S(φ)是φ∈ℬ的Schwarzian导数，该类似结论对ψ∈Σ同样成立。
英文摘要
We study the operator symbols of Dirichlet type introduced by Hedenmalm and Shimorin (2020), in connection with a given contraction on $L^2$ of the unit disk. They are always holomorphic functions on the bidisk. Such Dirichlet symbols associated with the Grunsky operator of a univalent function on the disk or exterior disk are of particular significance. From the work of Hedenmalm and Shimorin, we know they are characterized as solutions of a certain nonlinear wave equation. We perform a local analysis of such symbols near the diagonal on the bidisk, and in so doing, we provide alternative chart coordinates for the infinite-dimensional manifolds of univalent functions of the (exterior) disk. Those coordinates allow us to characterize $\logψ'$ for $ψ$ in the class $Σ$ of normalized univalent functions without explicitly touching the univalence property. Moreover, that manifold extends the universal Teichmüller space of Lipman Bers beyond the quasicircle boundary setting, allowing for even more fractality. The fractality of harmonic measure for the domain associated with the given univalent function can be studied in terms of the asymptotic variance introduced by McMullen (2008). The asymptotic variance captures the $L^2$ average amplitude of the nonlinearity. We here introduce the new concept of Schwarzian asymptotic variance, which measures the average amplitude of the Schwarzian derivative in place of the nonlinearity. For this new Schwarzian asymptotic variance, we find that the effective average amplitude of $(1-|z|^2)^4|\Sop(\vp)|^2$ on the disk in the hyperbolic metric sense is at most $9.07735\ldots$, considerably smaller than the maximum amplitude of $36$. Here, $\Sop(\vp)$ is the Schwarzian derivative of $φ\in\mathscr{S}$, and the analogous statement is valid for $ψ\inΣ$ as well.
Comments59 pages
