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On Removability Properties of \(\psi \)-Uniform Domains in Banach Spaces

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Abstract

Suppose that E is a real Banach space with dimension at least 2. The main aim of this paper is to show that a domain D in E is a \(\psi \)-uniform domain if and only if \(D\backslash P\) is a \(\psi _1\)-uniform domain, and D is a uniform domain if and only if \(D\backslash P\) also is a uniform domain, whenever P is a countable subset of D satisfying a quasihyperbolic separation condition. This condition requires that the quasihyperbolic distance with respect to D between each pair of distinct points in P has a lower bound greater than or equal to \(\frac{1}{2}\).

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Acknowledgments

This research was carried out when the first author was an academic visitor at the University of Turku and supported by the Academy of Finland grant of the second author with the Project Number 2600066611. The first and third authors are also grateful to the NSF of Guang-dong Province (No. 2014A030313471) and the Project of ISTCIPU in Guang-dong Province (No. 2014- KGJHZ007) for their support. The authors are indebted to the referee for the valuable suggestions.

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Authors and Affiliations

  1. Department of Mathematics, Hunan Normal University, Changsha, 410081, People’s Republic of China

    M. Huang

  2. Department of Mathematics and Statistics, University of Turku, 20014, Turku, Finland

    M. Vuorinen

  3. Department of Mathematics, Shantou University, Shantou, 515063, People’s Republic of China

    X. Wang

Authors
  1. M. Huang
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  2. M. Vuorinen
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  3. X. Wang
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Corresponding author

Correspondence to X. Wang.

Additional information

Communicated by Simeon Reich.

The research was partly supported by NSF of China (No. 11571216) and by the Academy of Finland, Project 2600066611.

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Huang, M., Vuorinen, M. & Wang, X. On Removability Properties of \(\psi \)-Uniform Domains in Banach Spaces. Complex Anal. Oper. Theory 11, 35–55 (2017). https://doi.org/10.1007/s11785-016-0566-z

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  • DOI: https://doi.org/10.1007/s11785-016-0566-z

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