{"title":"Douady–Earle Extensions of Circle Homeomorphisms with One-Point Differentiability at a Hölder Convergence Rate","authors":"Jinhua Fan, Jun Hu, Zhenyong Hu","doi":"10.1007/s40315-024-00540-9","DOIUrl":null,"url":null,"abstract":"<p>Let <i>h</i> be a sense-preserving homeomorphism of the unit circle <span>\\({\\mathbb {S}}\\)</span> and <span>\\(\\Phi (h)\\)</span> the Douady–Earle extension of <i>h</i> to the closure of the open disk <span>\\({\\mathbb {D}}\\)</span>. In this paper, assuming that <i>h</i> is differentiable at a point <span>\\(\\xi \\in {\\mathbb {S}}\\)</span> with <span>\\(\\alpha \\)</span>-Hölder convergence rate for some <span>\\(0<\\alpha <1\\)</span>, we prove a similar regularity for <span>\\(\\Phi (h)\\)</span> near <span>\\(\\xi \\)</span> on <span>\\({\\mathbb {D}}\\)</span> in any non-tangential direction towards <span>\\(\\xi \\)</span>.</p>","PeriodicalId":49088,"journal":{"name":"Computational Methods and Function Theory","volume":"5 1","pages":""},"PeriodicalIF":0.6000,"publicationDate":"2024-04-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Computational Methods and Function Theory","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s40315-024-00540-9","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
Let h be a sense-preserving homeomorphism of the unit circle \({\mathbb {S}}\) and \(\Phi (h)\) the Douady–Earle extension of h to the closure of the open disk \({\mathbb {D}}\). In this paper, assuming that h is differentiable at a point \(\xi \in {\mathbb {S}}\) with \(\alpha \)-Hölder convergence rate for some \(0<\alpha <1\), we prove a similar regularity for \(\Phi (h)\) near \(\xi \) on \({\mathbb {D}}\) in any non-tangential direction towards \(\xi \).
期刊介绍:
CMFT is an international mathematics journal which publishes carefully selected original research papers in complex analysis (in a broad sense), and on applications or computational methods related to complex analysis. Survey articles of high standard and current interest can be considered for publication as well.