Hostname: page-component-7b7567c8c-tnzcf Total loading time: 0 Render date: 2026-11-14T11:40:36.163Z Has data issue: false hasContentIssue false
  • English
  • Français

Φ-Bounded Harmonic Functions and the Classification of Harmonic Spaces

Published online by Cambridge University Press:  22 January 2016

Wellington H. Ow
Wellington H. Ow*
Affiliation:
Michigan State University
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the 'Save PDF' action button.

By a harmonic space we mean a pair (X, H) where X is a locally compact, non-compact, connected, locally connected Hausdorff space ; and H is a sheaf of harmonic functions defined as follows: Suppose to each open set Ω ⊂ X there corresponds a linear space H(Ω) of finitely-continuous real-valued functions defined on Ω. Then H = {H(Ω)}Ω must satisfy the three axioms of Brelot (1) and in addition Axiom 4 of Loeb (4): 1 is H-superharmonic in X.

Information

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 48 , December 1972, pp. 57 - 65
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1972

References

[1] Brelot, M., Lectures on potential theory, Tata Inst. Fund. Res., Bombay, 1960.Google Scholar
[2] Chow, K. and Glasner, M., Bounded in the mean solutions of Δu = Pu on Riemannian manifolds, Proc. Amer. Math. Soc. 26 (1970), 261–265.Google Scholar
[3] Constantinescu, C. and Cornea, y, Compactifications of harmonic spaces, Nagoya Math. J. 25 (1965), 1–57.CrossRefGoogle Scholar
[4] Loeb, P. A., An axiomatic treatment of pairs of elliptic differential equations, Ann. Inst. Fourier, Grenoble 16 (1966), 167–208.Google Scholar
[5] Loeb, P. A. and Walsh, B., A maximal regular boundary for solutions of elliptic differential equations, Ann. Inst. Fourier, Grenoble 18 (1968), 283–308.Google Scholar
[6] Nakai, M., Φ-bounded harmonic functions and classification of Riemann surfaces, Pacific J. Math. 15 (1965), 1329-1335.Google Scholar
[7] Nakai, M., On Φ-bounded harmonic functions, Ann. Inst. Fourier 16 (1966), 145–157.CrossRefGoogle Scholar
[8] Ow, W. H., Wiener’s compactification and Φ-bounded harmonic functions in the classification of harmonic spaces, Pacific J. Math, (to appear).Google Scholar
[9] Parreau, M., Sur les moyennes des fonctions harmoniques et analytiques et la classification des surfaces de Riemann, Ann. Inst. Fourier (Grenoble) 3 (1951), 103–197.Google Scholar
[10] Sario, L. and Nakai, M., Classification theory of Riemann surfaces, Springer-Verlag, Berlin-Heidelberg-New York, 1970.Google Scholar
[11] Schiff, J. L., Φ-bounded solutions of Δu = Pu on a Riemann surface, (to appear).Google Scholar