×
Armitage, D. H.; Gardiner, Stephen J.; Netuka, I.

Separation of points by classes of harmonic functions. (English) Zbl 0788.31006

Math. Proc. Camb. Philos. Soc. 113, No. 3, 561-571 (1993).
Two subsets of \(\mathbb{R}^ n\) are called equivalent if one is the direct image of the other under (i) a linear fractional transformation if \(n=2\), (ii) an isometry if \(n \geq 3\). A domain \(D\) is called a Denjoy domain if \(\mathbb{R}^ n \backslash D \subseteq \mathbb{R}^{n-1} \times \{0\}\). The main result states that the class of all positive harmonic functions on a Green domain \(\Omega\) separates the points of \(\Omega\), if and only if \(\Omega\) is not equivalent to a Denjoy domain whose Martin compactification is homeomorphic to the Alexandroff compactification of \(\mathbb{R}^ n\). Those domains for which certain other classes of harmonic functions separate the points, are also characterized.
Reviewer: N.A.Watson (Christchurch)

MSC:

31B05 Harmonic, subharmonic, superharmonic functions in higher dimensions

Keywords:

Denjoy domain; Martin compactification; Alexandroff compactification
PDF BibTeX XML Cite
\textit{D. H. Armitage} et al., Math. Proc. Camb. Philos. Soc. 113, No. 3, 561--571 (1993; Zbl 0788.31006)
Full Text: DOI

References:

[1] Falconer, The Geometry of Fractal Sets (1985) · Zbl 0587.28004
[2] Hayman, Subharmonic Functions 1 (1976)
[3] Doob, Classical Potential Theory and its Probabilistic Counterpart (1984) · Zbl 0549.31001
[4] DOI: 10.1007/BF01432280 · Zbl 0227.31010
[5] DOI: 10.1007/BF02384681 · Zbl 0455.31009
[6] DOI: 10.2307/2323940 · Zbl 0742.30039
[7] DOI: 10.2307/2034663
[8] Bauer, Function Algebras, Proceedings of an International Symposium on Function Algebras, Tulane, 1965 pp 122– (1966)
[9] Ancona, Ann. Inst. Fourier (Grenoble) 29 pp 71– (1979) · Zbl 0589.31002
[10] Tanaka, Hokkaido Math. J. 6 pp 302– (1977) · Zbl 0399.30033
[11] Stein, Introduction to Fourier Analysis (1971) · Zbl 0232.42007
[12] DOI: 10.2307/2047548 · Zbl 0644.30026
[13] Leutwiler, Ann. A cad. Sci. Fenn. Ser. AlMath. 12 pp 3– (1987)
[14] DOI: 10.1112/plms/s3-21.4.614 · Zbl 0207.41603
[15] DOI: 10.1007/BF01113851 · Zbl 0163.14204
[16] Herron, Canad. Math. Bull. 32 pp 286– (1989) · Zbl 0646.30013
[17] DOI: 10.2996/kmj/1138037356 · Zbl 0631.31001
[18] Helms, Introduction to Potential Theory (1969) · Zbl 0188.17203
[19] Eremenko, Approximation by Solutions of Partial Differential Equations pp 61– (1992)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.