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