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The parabolicity of Brelot's harmonic spaces

Part of: Axiomatic potential theory Riemann surfaces

Published online by Cambridge University Press:  09 April 2009

Hideo Imai
Hideo Imai
Affiliation:
Department of Mathematics, Daido Institute of Technology, Takiharu, Minami, Nagoya 457, Japan
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Abstract

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The parabolicity of Brelot's harmonic spaces is characterized by the fact that every positive harmonic function is of minimal growth at the ideal boundary.

MSC classification

Secondary: 31D05: Axiomatic potential theory 30F15: Harmonic functions on Riemann surfaces
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 59 , Issue 1 , August 1995 , pp. 20 - 27
Copyright
Copyright © Australian Mathematical Society 1995

References

[1]Agmon, S., ‘On positivity and decay of solutions of second order elliptic equations on Riemannian manifolds’, in: Methods of functional analysis and theory of elliptic equations (ed. Greco, D.) (Liguori, Naples, 1982) pp. 19–52.Google Scholar
[2]Constantinescu, C. and Cornea, A., Potential theory on harmonic spaces (Springer, Berlin, 1972).CrossRefGoogle Scholar
[3]Hervé, R-M., ‘Recherches axiomatiques sur la théorie des fonctions surharmoniques et du potentiel’, Ann. Inst. Fourier (Grenoble) 12 (1962), 415–571.Google Scholar
[4]Lahtinen, A., ‘On the existence of singular solutions of Δu = Pu’, Ann. Acad. Sci. Fenn. Ser. A I Math. Dissertationes 546 (1973).Google Scholar
[5]Loeb, P., ‘An axiomatic treatment of pairs of elliptic differential equations’, Ann. Inst. Fourier (Grenoble) 16 (1966), 167–208.CrossRefGoogle Scholar
[6]Maeda, F-Y., ‘Dirichlet integrals on harmonic spaces’, Lecture Notes in Math. 803 (Springer, Berlin, 1980).CrossRefGoogle Scholar
[7]Murata, M., ‘Isolated singularities and positive solutions of elliptic equations in Rn’, preprint, Aarhus Universitet (1986).Google Scholar
[8]Murata, M., ‘Structure of positive solutions to (–Δ + V)u = 0 in R n’, Duke Math. J. 53 (1986), 869–943.CrossRefGoogle Scholar
[9]Nakai, M. and Tada, T., ‘Picard dimension of rotation free signed densities’, Bull. Nagoya Inst. Tech. 43 (1991), 137–151 (in Japanese).Google Scholar
[10]Pinchover, Y., ‘On positive solutions of second-order elliptic equations, stability results, and classification’, Duke Math. J. 57 (1988), 955–980.CrossRefGoogle Scholar
[11]Sario, L. and Nakai, M., Classification theory of Riemann surfaces (Springer, Berlin, 1970).CrossRefGoogle Scholar