## 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.## References

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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