On Quasi-Invariance of Harmonic Measure and Hayman–Wu Theorem

IF 0.5 Q3 MATHEMATICS

Russian Mathematics Pub Date : 2024-06-04 DOI:10.3103/s1066369x24700087

S. Yu. Graf

引用次数: 0

Abstract

This study defines and describes the properties of the class of diffeomorphisms of the unit disk \(\mathbb{D} = \{ z\,:\;|{\kern 1pt} z{\kern 1pt} |\; < 1\} \) on the complex plane \(\mathbb{C}\) for which the harmonic measure of the boundary arcs of the slit disk has a limited distortion (i.e., is quasi-invariant). Estimates for derivative mappings of this class are obtained. We prove that such mappings are quasiconformal and quasi-isometries with respect to the pseudohyperbolic metric. An example of a mapping with the specified property is given. As an application, a generalization of the Hayman–Wu theorem to this class of such mappings is proved.

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论谐波量的准不变量和海曼-吴定理

摘要本研究定义并描述了复平面(\mathbb{C}\)上单位圆盘(\mathbb{D} = \{ z\,:\;|{\kern 1pt} z{\kern 1pt} |\; < 1\} \)的差分映射类的性质，对于这类映射，狭缝圆盘的边界弧的谐波度量具有有限的扭曲（即准不变）。我们得到了该类导数映射的估计值。我们证明了这类映射相对于伪双曲度量是类共形和类等距的。我们给出了一个具有上述性质的映射实例。作为应用，我们还证明了海曼-吴（Hayman-Wu）定理对这类映射的推广。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Russian Mathematics MATHEMATICS-

CiteScore

0.90

自引率

25.00%

发文量

期刊介绍： Russian Mathematics is a peer reviewed periodical that encompasses the most significant research in both pure and applied mathematics.