{"title":"On Quasi-Invariance of Harmonic Measure and Hayman–Wu Theorem","authors":"S. Yu. Graf","doi":"10.3103/s1066369x24700087","DOIUrl":null,"url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p>This study defines and describes the properties of the class of diffeomorphisms of the unit disk <span>\\(\\mathbb{D} = \\{ z\\,:\\;|{\\kern 1pt} z{\\kern 1pt} |\\; < 1\\} \\)</span> on the complex plane <span>\\(\\mathbb{C}\\)</span> 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.</p>","PeriodicalId":46110,"journal":{"name":"Russian Mathematics","volume":"26 1","pages":""},"PeriodicalIF":0.5000,"publicationDate":"2024-06-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Russian Mathematics","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.3103/s1066369x24700087","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
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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