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Uniform, Sobolev extension and quasiconformal circle domains

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Abstract

This paper contributes to the theory of uniform domains and Sobolev extension domains. We present new features of these domains and exhibit numerous relations among them. We examine two types of Sobolev extension domains, demonstrate their equivalence for bounded domains and generalize known sufficient geometric conditions for them. We observe that in the plane essentially all of these domains possess the trait that there is a quasiconformal self-homeomorphism of the extended plane which maps a given domain conformally onto a circle domain. We establish a geometric condition enjoyed by these plane domains which characterizes them among all quasicircle domains having no large and no small boundary components.

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Authors and Affiliations

  1. Department of Mathematics, University of Cincinnati, 45221-0025, Cincinnati, OH, USA

    David A. Herron

  2. Department of Mathematics, University of Jyväskylä, 40100, Jyväskylä, Finland

    Pekka Koskela

Authors
  1. David A. Herron
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  2. Pekka Koskela
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Partially supported by the University of Cincinnati’s Taft Foundation.

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Herron, D.A., Koskela, P. Uniform, Sobolev extension and quasiconformal circle domains. J. Anal. Math. 57, 172–202 (1991). https://doi.org/10.1007/BF03041069

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  • DOI: https://doi.org/10.1007/BF03041069

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