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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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A characterization of the unit disk and the harmonic measure doubling condition
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by Nikolaos Karamanlis PDF
Proc. Amer. Math. Soc. 147 (2019), 1671-1675 Request permission

Abstract:

Suppose $\Omega \subset \mathbb {C}$ is a bounded Jordan domain. Let $\Omega ^*=\overline {\mathbb {C}}\setminus \overline {\Omega }$ denote its complementary domain in the extended plane. A well-known theorem by Jerison and Kenig states that $\partial \Omega$ is a quasicircle if and only if both $\Omega$ and $\Omega ^*$ are doubling domains with respect to the harmonic measure. This theorem fails if we only assume that $\Omega$ is a doubling domain. We show that if $\Omega$ is a doubling domain with constant $c=1$, then it must be a disk.
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Additional Information
  • Nikolaos Karamanlis
  • Affiliation: Department of Mathematics, Aristotle University of Thessaloniki, 54124 Thessaloniki, Greece
  • Email: nikaraman@math.auth.gr
  • Received by editor(s): July 17, 2018
  • Received by editor(s) in revised form: August 30, 2018
  • Published electronically: January 9, 2019
  • Communicated by: Jeremy Tyson
  • © Copyright 2019 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 147 (2019), 1671-1675
  • MSC (2010): Primary 30C85, 30C62
  • DOI: https://doi.org/10.1090/proc/14371
  • MathSciNet review: 3910431