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On the Non-Minimal Martin Boundary Points

Published online by Cambridge University Press:  22 January 2016

Teruo Ikegami
Teruo Ikegami*
Affiliation:
Osaka City University
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In a Green space Ω we can introduce Martin’s topology and make it the Martin space Ω, Ω is a dense open subset of and the kernel

can be extended continuously to , where G(p, x) is a Green function in Ω and y0 the fixed point of Ω. is a metric space. is divided into two disjoint subsets Δ0, Δ1 and s ∊ Δ1 is characterized by the fact that K(s, x) is a minimal positive harmonic function in x∊Ω.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 29 , March 1967 , pp. 287 - 290
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1967

References

1) Brelot, M., Choquet, G., Espaces et lignes de Green. Annales Inst. Fourier 3 (1951), pp. 199–263.CrossRefGoogle Scholar

2) Brelot, M., Le problème de Dirichlet. Axiomatique et frontière de Martin. Journal de Math. 35 (1956), pp. 297–335 (pp. 329–330)Google Scholar, Cf. also Martin, R. S., Minimal positive harmonie functions, Trans. Amer. Math. Soc., 49 (1941), pp. 137–172 CrossRefGoogle Scholar. Parreau, M., Sur les moyennes des fonctions harmoniques et analytiques et la classification des surfaces de Riemann, Annales Inst. Fourier 3 (1952), pp. 103–197 CrossRefGoogle Scholar. Naïm, L., Sur le rôle de la frontière de R. S. Martin dans la théorie du potentiel, Annales Inst. Fourier 7 (1957), pp. 183–281.CrossRefGoogle Scholar

3) R. S. Martin, loc. cit., p. 137.

4) R. S. Martin, loc. cit., p. 157.

5) L. Nairn, loc. cit., p. 203 (théorème 3) and p. 205 (théorème 5).

6) We denote , where the metric dist (x, z 0) is the Martin’s metric.