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
DOI:
https://doi.org/10.1090/spmj/1420
Published electronically:
July 26, 2016
MathSciNet review:
3582947
Full-text PDF Free Access
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
Keywords:
Pseudocontinuation,
conformal mapping,
Hölder 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