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Harmonic morphisms applied to classical potential theory

Published online by Cambridge University Press:  11 January 2016

Bent Fuglede
Bent Fuglede*
Affiliation:
Department of Mathematical Sciences, University of Copenhagen, 2200 Copenhagen Ø, Denmarkfuglede@math.ku.dk
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Abstract

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It is shown that if ϕ denotes a harmonic morphism of type Bl between suitable Brelot harmonic spaces X and Y, then a function f, defined on an open set V ⊂ Y, is superharmonic if and only if f ∘ ϕ is superharmonic on ϕ–1(V) ⊂ X. The “only if” part is due to Constantinescu and Cornea, with ϕ denoting any harmonic morphism between two Brelot spaces. A similar result is obtained for finely superharmonic functions defined on finely open sets. These results apply, for example, to the case where ϕ is the projection from ℝN to ℝn (N > n ≥ 1) or where ϕ is the radial projection from ℝN \ {0} to the unit sphere in ℝN (N ≥ 2).

Keywords

31B0531C0531C1231C40
Type
Research Article
Information
Nagoya Mathematical Journal , Volume 202 , June 2011 , pp. 107 - 126
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 2011

References

[AG] Armitage, D. H. and Gardiner, S. J., Classical Potential Theory, Springer, Berlin, 2001.Google Scholar
[BW] Baird, P. and Wood, J. C., Harmonic Morphisms Between Riemannian Manifolds, Clarendon Press, Oxford, 2003.Google Scholar
[Ba] Bauer, H., Harmonische Räume und ihre Potentialtheorie, Lecture Notes in Math. 22, Springer, Berlin, 1966.Google Scholar
[BH] Bliedtner, J. and Hansen, W., Potential Theory — An Analytic and Probabilistic Approach to Balayage, Springer, Berlin, 1986.Google Scholar
[Br] Brelot, M., Lectures on Potential Theory, Tata Inst. Fund. Res., Mumbai, 1960.Google Scholar
[CC1] Constantinescu, C. and Cornea, A., Ideale Ränder Riemannscher Flächen, Ergeb. Math. Grenzgeb. 32, Springer, Berlin, 1963.Google Scholar
[CC2] Constantinescu, C. and Cornea, A., Compactifications of harmonic spaces, Nagoya Math. J. 25 (1965), 1–57.CrossRefGoogle Scholar
[CC3] Constantinescu, C. and Cornea, A., Potential Theory on Harmonic Spaces, Grundlehren Math. Wiss., Band 158, Springer, Berlin, 1972.Google Scholar
[DL] Deny, J. and Lelong, P., Étude des fonctions sousharmoniques dans un cylindre ou dans un cône, Bull. Soc. Math. France 75 (1947), 89–112.Google Scholar
[Do] Doob, J. L., Applications to analysis of a topological definition of smallness of a set, Bull. Amer. Math. Soc. (N.S.) 72 (1966), 579–600.Google Scholar
[F1] Fuglede, B., Connexion en topologie fine et balayage des mesures, Ann. Inst. Fourier (Grenoble) 21 (1971), 227–244.Google Scholar
[F2] Fuglede, B., Finely Harmonic Functions, Lecture Notes in Math. 289, Springer, Berlin, 1972.Google Scholar
[F3] Fuglede, B., Finely harmonic mappings and finely holomorphic functions, Ann. Acad. Sci. Fenn. Math. 2 (1976), 113–127.Google Scholar
[F4] Fuglede, B., Harmonic morphisms between Riemannian manifolds, Ann. Inst. Fourier (Grenoble) 28 (1978), 107–144.CrossRefGoogle Scholar
[F5] Fuglede, B., “Harmonic morphisms” in Complex Analysis (Joensuu, 1978), Lecture Notes in Math. 747, Springer, Berlin, 1979, 123–131.Google Scholar
[F6] Fuglede, B., Harnack sets and openness of harmonic morphisms, Math. Ann. 241 (1979), 181–186.Google Scholar
[G1] Gardiner, S. J., The Lusin-Primalov theorem for subharmonic functions, Proc. Amer. Math. Soc. 124 (1996), 3721–3727.Google Scholar
[G2] Gardiner, S. J., Growth properties of superharmonic functions along rays, Proc. Amer. Math. Soc. 128 (2000), 1963–1970.CrossRefGoogle Scholar
[GH] Gardiner, S. J. and Hansen, W., The Riesz decomposition of finely superharmonic functions, Adv. Math. 214 (2007), 417–436.Google Scholar
[Ha] Hansen, W., Abbildungen harmonischer Räume mit Anwendung auf die Laplace und Wärmeleitungsgleichung, Ann. Inst. Fourier (Grenoble) 21 (1971), 203–216.Google Scholar
[HKM] Heinonen, J., Kilpeläinen, T., and Martio, O., Harmonic morphisms in nonlinear potential theory, Nagoya Math. J. 125 (1992), 115–140.CrossRefGoogle Scholar
[Hs] Heins, M., On the Lindelöf principle, Ann. of Math. (2) 61 (1955), 440–473.Google Scholar
[He] Hervé, R.-M., Recherches axiomatiques sur la théorie des fonctions surharmoniques et du potentiel, Ann. Inst. Fourier (Grenoble) 12 (1962), 415–571.Google Scholar
[Ish] Ishihara, T., A mapping of Riemannian manifolds which preserves harmonic functions, J. Math. Kyoto Univ. 19 (1979), 215–229.Google Scholar
[Jac] Jacobi, C. G. J., Über eine Lösung der partiellen Differentialgleichung , J. Reine Angew. Math. 36 (1848), 113–134.Google Scholar
[Jan] Janssen, K., A cofine domination principle for harmonic spaces, Math. Z. 141 (1975), 185–191.Google Scholar
[La1] Laine, I., Covering properties of harmonic Bl-mappings, II, Ann. Acad. Sci. Fenn. Math. 570 (1974), 3–13.Google Scholar
[La2] Laine, I., Covering properties of harmonic Bl-mappings, III, Ann. Acad. Sci. Fenn. Math. 1 (1975), 309–325.Google Scholar
[Me] Meghea, C., Compactification des espaces harmoniques, Lecture Notes in Math. 222, Springer, Berlin, 1971.Google Scholar