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Equicontinuity on Harmonic Spaces

Published online by Cambridge University Press:  22 January 2016

Corneliu Constantinescu
Corneliu Constantinescu*
Affiliation:
Institutul de Matematică, str. M. Eminescit 47 Bucureşti 3, Rumania
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G. Mokobodzki proved [5] that on any harmonic space with countable basis satisfying the axioms 1, 2, T+, K0 [2] [1] any equally bounded set of harmonic functions is equicontinuous. P. Loeb and B. Walsh showed [4] that the same property holds on a harmonic space without countable basis, if Brelot’s axiom 3 is fulfilled. The aim of this paper is to generalize these results to a harmonic space X satisfying only the axioms 1, 20, K1, [2] [1] where 20 is a weakened form of axiom 2. As a corollary we get: if any point of X possesses two open neighbourhoods U, V such that the set of harmonic functions on U separates the points of U∩V, then X has locally a countable basis.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 29 , March 1967 , pp. 1 - 6
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1967

References

[1] Bauer, H., Axiomatische Behandlung des Dirichletschen Problems für elliptische und parabolische Differentialgleichungen, Math. Ann. 146 (1962), 1–59.CrossRefGoogle Scholar
[2] Brelot, M., Lectures on Potential Theory (part. IV), Tata Institute of Fundamental Research, Bombay, (1960).Google Scholar
[3] Köthe, G., Topologische lineare Räume, (zweite Auflage) Springer Verlag, Berlin-Heidelberg-New York (1966).CrossRefGoogle Scholar
[4] Loeb, P., Walsh, B., The Equivalence of Harnack’s Principle and Harnack’s Inequality in the Axiomatic System of Brelot, Ann. Inst. Fourier, 15, 2 (1965), 597–600.CrossRefGoogle Scholar
[5] Mokobodzki, G., Représentations intégrales des fonctions harmoniques et surharmoniques (1964), unpublished.Google Scholar