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Smoothness of a conformal mapping on a subset of the boundary


Author: N. A. Shirokov
Translated by: S. Kislyakov
Original publication: Algebra i Analiz, tom 27 (2015), nomer 5.
Journal: St. Petersburg Math. J. 27 (2016), 841-849
MSC (2010): Primary 30C35
Published electronically: July 26, 2016
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Abstract | References | Similar Articles | Additional Information

Abstract: A conformal mapping $ f$ of the unit disk onto a Jordan domain $ G$ is considered. The boundary of $ G$ has the following structure. Another Jordan domain $ H$ is fixed whose boundary has Hölder smoothness $ a>1$, and a countable family of open arcs dense in the boundary is specified. $ G$ is obtained by replacement of each of these distinguished arcs with a Hölder arc of smoothness $ b$, $ 1<b<a$, having the same end-points. Thus, $ G$ has Hölder smoothness $ b$. It is shown that if the lengths of the distinguished arcs decay sufficiently fast (depending on $ a$ and $ b$), the function $ f$ still has Hölder smoothness $ a$ on a set of positive measure on the unit circle. The numbers $ a$ and $ b$ are assumed to be nonintegers.


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Additional Information

N. A. Shirokov
Affiliation: Department of Mathematics and Mechanics, St. Petersburg State University, Universitetskiĭ pr. 28, Petrodvorets, St. Petersburg 198504, Russia
Email: nikolai.shirokov@gmail.com

DOI: https://doi.org/10.1090/spmj/1420
Keywords: Pseudocontinuation, conformal mapping, H\"older classes
Received by editor(s): December 15, 2014
Published electronically: July 26, 2016
Additional Notes: Supported by RFBR (grant no. 14-01-00198)
Article copyright: © Copyright 2016 American Mathematical Society