Hostname: page-component-8448b6f56d-t5pn6 Total loading time: 0 Render date: 2024-04-24T23:48:17.339Z Has data issue: false hasContentIssue false

Separation of points by classes of harmonic functions

Published online by Cambridge University Press:  24 October 2008

D. H. Armitage
Affiliation:
Department of Pure Mathematics, Queen's University, Belfast BT7 1NN, Northern Ireland
S. J. Gardiner*
Affiliation:
Department of Mathematics & Statistics, McGill University, Montreal, Quebec, Canada, H3A 2K6
I. Netuka
Affiliation:
Mathematical Institute, Charles University, Sokolovská 83, CS-186 00 Praha, Czech Republic
*
Permanent address: Department of Mathematics, University College, Dublin 4, Ireland

Abstract

We characterize those domains Ω in ℝN for which the positive harmonic functions on Ω separate the points of Ω.

Type
Research Article
Copyright
Copyright © Cambridge Philosophical Society 1993

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Ancona, A.. Une propriété de la compactification de Martin d'un domaine Euclidien. Ann. Inst. Fourier (Grenoble) 29 (1979), 7190.CrossRefGoogle Scholar
[2]Bauer, H.. Aspects of linearity in the theory of function algebras. In Function Algebras, Proceedings of an International Symposium on Function Algebras, Tulane, 1965 (Scott, Foresman and Co., 1966), pp. 122137.Google Scholar
[3]Bear, H. S.. A geometric characterization of Gleason parts. Proc. Amer. Math. Soc. 16 (1965), 407412.CrossRefGoogle Scholar
[4]Bear, H. S.. Part metric and hyperbolic metric. Amer. Math. Monthly 98 (1991), 109123.CrossRefGoogle Scholar
[5]Benedicks, M.. Positive harmonic functions vanishing on the boundary of certain domains in Rn. Ark. Mat. 18 (1980), 5371.CrossRefGoogle Scholar
[6]Bliedtner, J. and Janssen, K.. Harnacksche Kegel und Metrik in harmonischen Räumen. Math. Ann. 198 (1972), 8597.CrossRefGoogle Scholar
[7]Doob, J. L.. Classical Potential Theory and its Probabilistic Counterpart (Springer-Verlag, 1984).CrossRefGoogle Scholar
[8]Eremenko, A. E. and Lyons, T. J.. Finely open sets in the limit set of a geometrically finite Kleinian group. In Approximation by Solutions of Partial Differential Equations (editors Fuglede, B. et al. ) (Kluwer, 1992), pp. 6167.CrossRefGoogle Scholar
[9]Falconer, K. J.. The Geometry of Fractal Sets (Cambridge University Press, 1985).CrossRefGoogle Scholar
[10]Hayman, W. K. and Kennedy, P. B.. Subharmonic Functions, vol. 1 (Academic Press, 1976).Google Scholar
[11]Helms, L. L.. Introduction to Potential Theory (Wiley, 1969).Google Scholar
[12]Herron, D. A.. The Harnack and other conformally invariant metrics. Kodai Math. J. 10 (1987), 919.CrossRefGoogle Scholar
[13]Herron, D. A. and Schiff, J. L.. Positive harmonic functions and complete metrics. Canad. Math. Bull. 32 (1989), 286297.CrossRefGoogle Scholar
[14]Köhn, J.. Die Harnacksehe Metrik in der Theorie der harmonischen Funktionen. Math. Z. 91 (1966), 5064.CrossRefGoogle Scholar
[15]Küran, Ü.. Study of superharmonic functions in Rn × (0, + ∞) by a passage to Rn+3. Proc. London Math. Soc. (3) 21 (1970), 614636.Google Scholar
[16]Leutwiler, H.. On a distance invariant under Möbius transformations in RN. Ann. A cad. Sci. Fenn. Ser. AlMath. 12 (1987), 317.Google Scholar
[17]Segawa, S.. Martin boundaries and Denjoy domains. Proc. Amer. Math. Soc. 103 (1988), 177183.CrossRefGoogle Scholar
[18]Stein, E. M. and Weiss, G.. Introduction to Fourier Analysis (Princeton University Press, 1971).Google Scholar
[19]Tanaka, H.. On Harnack's pseudo-distance. Hokkaido Math. J. 6 (1977), 302305.CrossRefGoogle Scholar