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St. Petersburg Mathematical Journal

ISSN 1547-7371(online) ISSN 1061-0022(print)



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: 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.

References [Enhancements On Off] (What's this?)

  • 1. Ch. Pommerenke, Boundary behaviour of conformal maps, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 299, Springer-Verlag, Berlin, 1992. MR 1217706
  • 2. E. M. Dyn′kin, Pseudoanalytic continuation of smooth functions. Uniform scale, Mathematical programming and related questions (Proc. Seventh Winter School, Drogobych, 1974) Central Èkonom.-Mat. Inst. Akad. Nauk SSSR, Moscow, 1976, pp. 40–73 (Russian). MR 0587795
  • 3. E. M. Dyn′kin, The pseudoanalytic extension, J. Anal. Math. 60 (1993), 45–70. MR 1253229,
  • 4. E. M. Dyn′kin, Nonanalytic symmetry principle and conformal mappings, Algebra i Analiz 5 (1993), no. 3, 119–142 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 5 (1994), no. 3, 523–544. MR 1239901

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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

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

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