## Surfaces carrying sufficiently many Dirichlet finite harmonic functions that are automatically bounded.(English)Zbl 1264.30027

Let $$H(R)$$ be the linear space of harmonic functions on a Riemann surface $$R$$. Let $$HB(R)$$ and $$HD(R)$$ be the linear subspaces of $$H(R)$$ consisting of the bounded harmonic functions on $$R$$ and the harmonic functions on $$R$$ with finite Dirichlet integral, respectively. In general, there is no inclusion relation between these two subspaces, but there are a lot of Riemann surfaces satisfying $$HB(R)\subset HD(R)$$ with the linear dimension $$\dim HB(R)<\infty$$ or $$HD(R)\subset HB(R)$$ with $$\dim HD(R)<\infty$$. In this paper the author studies the problem whether the relation $$HD(R)\subset HB(R)$$ always implies the finiteness of $$\dim HD(R)$$ or not, and shows the existence of a Riemann surface $$R$$ such that $$HD(R)\subset HB(R)$$ with $$\dim HD(R)=\infty$$. For the construction of such a Riemann surface, the author makes use of the Sario-Tôki disc. With respect to the converse inclusion relation, it is already shown by the author that there exists a Riemann surface $$R$$ such that $$HB(R)\subset HD(R)$$ with $$\dim HB(R)=\infty$$.

### MSC:

 30F20 Classification theory of Riemann surfaces 30F25 Ideal boundary theory for Riemann surfaces 30F15 Harmonic functions on Riemann surfaces 31A05 Harmonic, subharmonic, superharmonic functions in two dimensions
Full Text:

### References:

 [1] L. V. Ahlfors and L. Sario, Riemann Surfaces, Princeton Math. Ser., 26 , Princeton Univ. Press, 1960. · Zbl 0196.33801 [2] C. Constantinescu und A. Cornea, Ideale Ränder Riemannscher Flächen, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 32 , Springer-Verlag, 1963. · Zbl 0112.30801 [3] J. Heinonen, T. Kilpeläinen and O. Martio, Nonlinear Potential Theory of Degenerate Elliptic Equations, Oxford Mathematical Monographs, Oxford University Press, 1993. · Zbl 0780.31001 [4] F.-Y. Maeda, Dirichlet Integrals on Harmonic Spaces, Lecture Notes in Math., 803 , Springer-Verlag, 1980. · Zbl 0426.31001 [5] H. Masaoka, The class of harmonic functions with finite Dirichlet integrals and the harmonic Hardy spaces on a hyperbolic Riemann surface, RIMS Kôkyûroku, 1669 (2009), 81-90. [6] M. Nakai, Extremal functions for capacities, Proceedings of the Workshop on Potential Theory, Hiroshima, 2007, pp.,83-102. [7] M. Nakai, Extremal functions for capacities, J. Math. Soc. Japan, 61 (2009), 345-361. · Zbl 1185.31002 [8] M. Nakai, Spectral resolutions of bounded harmonic functions, Proceeding of the Workshop on Potential Theory in Akita, 2008, pp.,81-104. [9] K. Oikawa, Welding of polygons and the type of Riemann surfaces, Kōdai Math. Sem. Rep., 13 (1961), 37-52. · Zbl 0129.05702 [10] K. Oikawa, Riemann Surfaces, Kyoritu Lecture Series, 22 , Kyoritu Publishing Co., 1987. [11] L. Sario and M. Nakai, Classification Theory of Riemann Surfaces, Grund-lehren der mathematischen Wissenschaften in Einzelldarstellungen, Band 164 , Springer-Verlag, 1970. · Zbl 0199.40603 [12] L. Sario, M. Nakai, C. Wang and L. O. Chung, Classification Theory of Riemannian Manifolds, Lecture Notes in Math., 803 , Springer-Verlag, 1980. · Zbl 0355.31001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.