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Harmonic morphisms applied to classical potential theory. (English) Zbl 1235.31002
Let $$X$$ and $$Y$$ be Brelot harmonic spaces. A continuous map $$\phi:X\rightarrow Y$$ is called a harmonic morphism if, for any open subset $$V$$ of $$Y$$ and any harmonic function $$f$$ on $$V$$, the function $$f\circ \phi$$ is harmonic on $$\phi ^{-1}(V)$$. Further, a harmonic morphism $$\phi$$ is said to be of type Bl if, for any open $$V\subset Y$$ and any locally bounded potential $$p$$ on $$V$$, the function $$p\circ \phi$$ is a potential on $$\phi^{-1}(V)$$. The author investigates the interplay between harmonic morphisms and the fine topology, thinness, polarity, superharmonicity and fine superharmonicity (in the last two cases, Bl harmonic morphisms are considered). He then applies his general theory to two specific cases in classical potential theory, namely where $$\phi$$ is a projection from $$\mathbb{R}^{N}$$ to a lower dimensional space $$\mathbb{R}^{n}$$, and where $$\phi$$ is the radial projection from $$\mathbb{R}^{N}\backslash \{0\}$$ to the unit sphere. This leads to the recovery and generalization of results of several authors. The paper concludes with a consideration of the more general notion of a finely harmonic morphism.

##### MSC:
 31B05 Harmonic, subharmonic, superharmonic functions in higher dimensions 31C05 Harmonic, subharmonic, superharmonic functions on other spaces 31C12 Potential theory on Riemannian manifolds and other spaces 31C40 Fine potential theory; fine properties of sets and functions
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##### References:
 [1] D. H. Armitage and S. J. Gardiner, Classical Potential Theory , Springer, Berlin, 2001. · Zbl 0972.31001 [2] P. Baird and J. C. Wood, Harmonic Morphisms Between Riemannian Manifolds , Clarendon Press, Oxford, 2003. · Zbl 1055.53049 [3] H. Bauer, Harmonische Räume und ihre Potentialtheorie , Lecture Notes in Math. 22 , Springer, Berlin, 1966. [4] J. Bliedtner and W. Hansen, Potential Theory - An Analytic and Probabilistic Approach to Balayage , Springer, Berlin, 1986. · Zbl 0706.31001 [5] M. Brelot, Lectures on Potential Theory , Tata Inst. Fund. Res., Mumbai, 1960. · Zbl 0098.06903 [6] C. Constantinescu and A. Cornea, Ideale Ränder Riemannscher Flächen , Ergeb. Math. Grenzgeb. 32 , Springer, Berlin, 1963. · Zbl 0112.30801 [7] C. Constantinescu and A. Cornea, Compactifications of harmonic spaces , Nagoya Math. J. 25 (1965), 1-57. · Zbl 0138.36701 [8] C. Constantinescu and A. Cornea, Potential Theory on Harmonic Spaces , Grundlehren Math. Wiss., Band 158, Springer, Berlin, 1972. · Zbl 0248.31011 [9] J. Deny and P. Lelong, Étude des fonctions sousharmoniques dans un cylindre ou dans un cône , Bull. Soc. Math. France 75 (1947), 89-112. · Zbl 0033.06401 [10] J. L. Doob, Applications to analysis of a topological definition of smallness of a set , Bull. Amer. Math. Soc. (N.S.) 72 (1966), 579-600. · Zbl 0142.09001 [11] B. Fuglede, Connexion en topologie fine et balayage des mesures , Ann. Inst. Fourier (Grenoble) 21 (1971), 227-244. · Zbl 0208.13802 [12] B. Fuglede, Finely Harmonic Functions , Lecture Notes in Math. 289 , Springer, Berlin, 1972. · Zbl 0248.31010 [13] B. Fuglede, Finely harmonic mappings and finely holomorphic functions , Ann. Acad. Sci. Fenn. Math. 2 (1976), 113-127. · Zbl 0345.31008 [14] B. Fuglede, Harmonic morphisms between Riemannian manifolds , Ann. Inst. Fourier (Grenoble) 28 (1978), 107-144. · Zbl 0339.53026 [15] B. Fuglede, “Harmonic morphisms” in Complex Analysis (Joensuu, 1978) , Lecture Notes in Math. 747 , Springer, Berlin, 1979, 123-131. · Zbl 0414.53033 [16] B. Fuglede, Harnack sets and openness of harmonic morphisms , Math. Ann. 241 (1979), 181-186. · Zbl 0401.31004 [17] S. J. Gardiner, The Lusin-Primalov theorem for subharmonic functions , Proc. Amer. Math. Soc. 124 (1996), 3721-3727. JSTOR: · Zbl 0868.31006 [18] S. J. Gardiner, Growth properties of superharmonic functions along rays , Proc. Amer. Math. Soc. 128 (2000), 1963-1970. JSTOR: · Zbl 0945.31002 [19] S. J. Gardiner and W. Hansen, The Riesz decomposition of finely superharmonic functions , Adv. Math. 214 (2007), 417-436. · Zbl 1147.31003 [20] W. Hansen, Abbildungen harmonischer Räume mit Anwendung auf die Laplace und Wärmeleitungsgleichung , Ann. Inst. Fourier (Grenoble) 21 (1971), 203-216. · Zbl 0208.13701 [21] J. Heinonen, T. Kilpeläinen, and O. Martio, Harmonic morphisms in nonlinear potential theory , Nagoya Math. J. 125 (1992), 115-140. · Zbl 0776.31007 [22] M. Heins, On the Lindelöf principle , Ann. of Math. (2) 61 (1955), 440-473. JSTOR: · Zbl 0065.31102 [23] R.-M. Hervé, Recherches axiomatiques sur la théorie des fonctions surharmoniques et du potentiel , Ann. Inst. Fourier (Grenoble) 12 (1962), 415-571. · Zbl 0101.08103 [24] T. Ishihara, A mapping of Riemannian manifolds which preserves harmonic functions , J. Math. Kyoto Univ. 19 (1979), 215-229. · Zbl 0421.31006 [25] C. G. J. Jacobi, Über eine Lösung der partiellen Differentialgleichung \partial 2 V / \partial x 2 + \partial 2 V / \partial y 2 + \partial 2 V / \partial z 2 =0, J. Reine Angew. Math. 36 (1848), 113-134. · ERAM 036.1002cj [26] K. Janssen, A cofine domination principle for harmonic spaces , Math. Z. 141 (1975), 185-191. · Zbl 0283.31008 [27] I. Laine, Covering properties of harmonic Bl -mappings, II , Ann. Acad. Sci. Fenn. Math. 570 (1974), 3-13. · Zbl 0294.31011 [28] I. Laine, Covering properties of harmonic Bl -mappings, III , Ann. Acad. Sci. Fenn. Math. 1 (1975), 309-325. · Zbl 0328.31014 [29] C. Meghea, Compactification des espaces harmoniques , Lecture Notes in Math. 222 , Springer, Berlin, 1971. · Zbl 0217.10601
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