Abstract
We consider multiply hyperharmonic functions on the product space of two harmonic spaces in the sense of Constantinescu and Cornea. Earlier multiply superharmonic and harmonic functions have been studied in Brelot spaces notably by GowriSankaran. Important examples of Brelot spaces are solutions of elliptic differential equations. The theory of general harmonic spaces covers in addition to Brelot spaces also solution of parabolic differential equations. A locally lower bounded function is multiply hyperharmonic on the product space of two harmonic spaces if it is a hyperharmonic function in each variable for every fixed value of the other. We prove similar results as in Brelot spaces, but our approach is different. We study sheaf properties of multiply hyperharmonic functions. Our main theorem states that multiply hyperharmonic functions are lower semicontinuous and satisfy the axiom of completeness with respect to products of relatively compact sets. We also study nearly multiply hyperharmonic functions.
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References
Avanissian, V.: Fonctions plusi-Sousharmoniques et fonctions doublement sousharmoniques, Ann. Sci. Ecole Norm. Sup. (3) 78(1961), 101-161.
Bauer, H.: Harmonische Räume und ihre Potentialtheorie, Lecture Notes in Math. 22, Springer-Verlag, Berlin-Heidelberg, New York 1966.
Constantinescu C. and Cornea, A.: Potential Theory on Harmonic Spaces, Springer-Verlag, Berlin-Heidelberg-New York, 1972.
GowriSankaran, K.: Iterated fine limits and non-tangential limits, Trans. Amer. Math. Soc. 173(1972), 71-92.
GowriSankaran, K.: Multiply harmonic functions, Nagoya Math. J. 28(1966), 27-48.
GowriSankaran, K.: On a problem of Doob concerning multiply superharmonic functions, Nagoya Math. J. 39(1970), 127-132.
GowriSankaran, K.: Integral representation for a class of multiply superharmonic functions, Ann. Inst. Fourier 23(4) (1973), 105-143.
GowriSankaran, K.: Multiply superharmonic functions, Ann. Inst. Fourier 25(3-4) (1975), 235- 244.
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Eriksson-Bique, SL. Multiply Hyperharmonic Functions. Potential Analysis 6, 149–162 (1997). https://doi.org/10.1023/A:1008619213286
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DOI: https://doi.org/10.1023/A:1008619213286