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A characterization of the unit disk and the harmonic measure doubling condition


Author: Nikolaos Karamanlis
Journal: Proc. Amer. Math. Soc. 147 (2019), 1671-1675
MSC (2010): Primary 30C85, 30C62
DOI: https://doi.org/10.1090/proc/14371
Published electronically: January 9, 2019
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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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Nikolaos Karamanlis
Affiliation: Department of Mathematics, Aristotle University of Thessaloniki, 54124 Thessaloniki, Greece
Email: nikaraman@math.auth.gr

DOI: https://doi.org/10.1090/proc/14371
Keywords: Harmonic measure, quasidisk
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
Article copyright: © Copyright 2019 American Mathematical Society