Proceedings of the Japan Academy, Series A, Mathematical Sciences

Nonseparability of Banach spaces of bounded harmonic functions on Riemann surfaces

Mitsuru Nakai

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Abstract

The separability of certain seminormed spaces of harmonic functions on Riemann surfaces will be considered. An application of the result obtained in the above to some inverse inclusion problem in the classification theory of Riemann surfaces will be appended.

Article information

Source
Proc. Japan Acad. Ser. A Math. Sci., Volume 87, Number 8 (2011), 130-135.

Dates
First available in Project Euclid: 3 October 2011

Permanent link to this document
https://projecteuclid.org/euclid.pja/1317647394

Digital Object Identifier
doi:10.3792/pjaa.87.130

Mathematical Reviews number (MathSciNet)
MR2777229

Zentralblatt MATH identifier
1254.30063

Subjects
Primary: 30F20: Classification theory of Riemann surfaces
Secondary: 30F25: Ideal boundary theory 30F15: Harmonic functions on Riemann surfaces 46B26: Nonseparable Banach spaces

Keywords
Banach space Dirichlet integral Green function harmonic function Hilbert space separable

Citation

Nakai, Mitsuru. Nonseparability of Banach spaces of bounded harmonic functions on Riemann surfaces. Proc. Japan Acad. Ser. A Math. Sci. 87 (2011), no. 8, 130--135. doi:10.3792/pjaa.87.130. https://projecteuclid.org/euclid.pja/1317647394


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References

  • L. V. Ahlfors and L. Sario, Riemann surfaces, Princeton Mathematical Series, No. 26, Princeton Univ. Press, Princeton, NJ, 1960.
  • C. Constantinescu und A. Cornea, Ideale Ränder Riemannscher Flächen, Ergebnisse der Mathematik und ihrer Grenzgebiete, N. F., Bd. 32, Springer, Berlin, 1963.
  • J. L. Doob, Boundary properties for functions with finite Dirichlet integrals, Ann. Inst. Fourier (Grenoble) 12 (1962), 573–621.
  • J. Douglas, Solution of the problem of Plateau, Trans. Amer. Math. Soc. 33 (1931), no. 1, 263–321.
  • N. Dunford and J. T. Schwartz, Linear Operators, (Part I: General Theory), Pure and Applied Mathematics, Vol. 7, Interscience Publishers, 1967.
  • F.-Y. Maeda, Dirichlet integrals on harmonic spaces, Lecture Notes in Mathematics, 803, Springer, Berlin, 1980.
  • H. Masaoka, The classes of bounded harmonic functions and harmonic functions with finite Dirichlet integrals on hyperbolic Riemann surfaces, RIMS Kôkyûroku (Lecture Notes Series at Research Inst. Kyoto Univ.), 1553 (2007), 132–136.
  • H. Masaoka, The class of harmonic functions with finite Dirichlet integrals and the harmonic Hardy spaces on a hyperbolic Riemann surface, RIMS Kôkyûroku (Lecture Notes Series at Research Inst. Kyoto Univ.), 1669 (2009), 81–90.
  • H. Masaoka, The classes of bounded harmonic functions and harmonic functions with finite Dirichlet integrals on hyperbolic Riemann surfaces, Kodai Math. J. 33 (2010), no. 2, 233–239.
  • M. Nakai, Extremal functions for capacities, in the Workshop on Potential Theory 2007 (Hiroshima, 2007), 83–102.
  • M. Nakai, Extremal functions for capacities, J. Math. Soc. Japan 61 (2009), no. 2, 345–361.
  • M. Nakai, Nonreflexivity of Banach spaces of bounded harmonic functions on Riemann surfaces, Proc. Japan Acad. Ser. A Math. Sci. 87 (2011), no. 1, 1–4.
  • M. Nakai and S. Segawa, Tôki covering surfaces and their applications, J. Math. Soc. Japan 30 (1978), no. 2, 359–373.
  • M. Nakai and T. Tada, Bounded harmonic functions, Bull. Daido Univ. 45 (2009), 9–13.
  • C. E. Rickart, General theory of Banach algebras, The University Series in Higher Mathematics, D. van Nostrand Co., Inc., Princeton, NJ, 1960.
  • L. Sario and M. Nakai, Classification theory of Riemann surfaces, Die Grundlehren der mathematischen Wissenschaften, Band 164, Springer, New York, 1970.
  • M. H. Stone, Boundedness properties in function-lattices, Canadian J. Math. 1 (1949), 176–186.
  • K. Yosida, Functional Analysis, Grundlehren der mathematischen Wissenschaften in Einzelldarstellungen, Band 123, Springer, Berlin, 1965.