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Proceedings of the American Mathematical Society

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



Uniform and Sobolev extension domains

Authors: David A. Herron and Pekka Koskela
Journal: Proc. Amer. Math. Soc. 114 (1992), 483-489
MSC: Primary 46E35; Secondary 30C65
MathSciNet review: 1075947
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Abstract: We prove that if a domain $ D \subset {{\mathbf{R}}^n}$ is quasiconformally equivalent to a uniform domain, then $ D$ is an extension domain for the Sobolev class $ W_n^1$ if and only if $ D$ is locally uniform. We provide examples which suggest that this result is best possible. We exhibit a list of equivalent conditions for domains quasiconformally equivalent to uniform domains, one of which characterizes the quasiconformal homeomorphisms between uniform and locally uniform domains.

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Keywords: Sobolev extension domain, uniform domain, locally uniform domain, quasiconformal homeomorphism, quasisymmetry
Article copyright: © Copyright 1992 American Mathematical Society