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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Non-tangential limits, fine limits and the Dirichlet integral
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by Stephen J. Gardiner PDF
Proc. Amer. Math. Soc. 129 (2001), 3379-3387 Request permission

Abstract:

Let $B$ denote the unit ball in $\mathbb {R}^{n}.$ This paper characterizes the subsets $E$ of $B$ with the property that $\sup _{E}h=\sup _{B}h$ for all harmonic functions $h$ on $B$ which have finite Dirichlet integral. It also examines, in the spirit of a celebrated paper of Brelot and Doob, the associated question of the connection between non-tangential and fine cluster sets of functions on $B$ at points of the boundary.
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Additional Information
  • Stephen J. Gardiner
  • Affiliation: Department of Mathematics, University College Dublin, Dublin 4, Ireland
  • MR Author ID: 71385
  • ORCID: 0000-0002-4207-8370
  • Email: stephen.gardiner@ucd.ie
  • Received by editor(s): December 17, 1999
  • Received by editor(s) in revised form: April 3, 2000
  • Published electronically: April 25, 2001
  • Communicated by: Albert Baernstein II
  • © Copyright 2001 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 129 (2001), 3379-3387
  • MSC (2000): Primary 31B25
  • DOI: https://doi.org/10.1090/S0002-9939-01-05952-4
  • MathSciNet review: 1845016