Tohoku Mathematical Journal

Square means versus Dirichlet integrals for harmonic functions on Riemann surfaces

Hiroaki Masaoka and Mitsuru Nakai

Full-text: Open access


We show rather unexpectedly and surprisingly the existence of a hyperbolic Riemann surface $W$ enjoying the following two properties: firstly, the converse of the celebrated Parreau inclusion relation that the harmonic Hardy space $HM_{2}(W)$ with exponent 2 consisting of square mean bounded harmonic functions on $W$ includes the space $HD(W)$ of Dirichlet finite harmonic functions on $W$, and a fortiori $HM_{2}(W)=HD(W)$, is valid; secondly, the linear dimension of $HM_{2}(W)$, hence also that of $HD(W)$, is infinite.

Article information

Tohoku Math. J. (2), Volume 64, Number 2 (2012), 233-259.

First available in Project Euclid: 2 July 2012

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 30F20: Classification theory of Riemann surfaces
Secondary: 30F25: Ideal boundary theory 30F15: Harmonic functions on Riemann surfaces 31A15: Potentials and capacity, harmonic measure, extremal length [See also 30C85]

Afforested surface Dirichlet finite Hardy space Joukowski coordinate mean bounded Parreau decomposition quasibounded singular Wiener harmonic boundary


Masaoka, Hiroaki; Nakai, Mitsuru. Square means versus Dirichlet integrals for harmonic functions on Riemann surfaces. Tohoku Math. J. (2) 64 (2012), no. 2, 233--259. doi:10.2748/tmj/1341249373.

Export citation


  • L. V. Ahlfors and L. Sario, Riemann surfaces, Princeton Mathematical Series, No. 26, Princeton Univ. Press, Princeton, N.J., 1960.
  • C. Constantinescu und A. Cornea, Ideale Ränder Riemannscher Flächen, Ergebnisse der Mathematik und ihre Grenzgebiete, Band 32, Springer-Verlag, 1963.
  • J. L. Doob, Boundary properties of functions with finite Dirichlet integrals, Ann. Inst. Fourier 12 (1962), 573–621.
  • F.-Y. Maeda, Dirichlet integrals on harmonic spaces, Lecture Notes in Mathematics 803, Springer-Verlag, Berlin, 1980.
  • 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.
  • H. Masaoka, The classes of bounded harmonic functions and harmonic functions with finite Dirichlet integrals on hyperbolic Riemann surfaces, Kodai Math. J. 33 (2010), 233–239.
  • M. Nakai, Extremal functions for capacities, Proceedings of the Workshop on Potential Theory 2007 in Hiroshima, 83–102.
  • M. Nakai, Extremal functions for capacities, J. Math. Soc. Japan 61 (2009), 345–361.
  • M. Nakai, Nonreflexivity of Banach spaces of bounded harmonic functions on Riemann surfaces, Proc. Japan Acad. Ser. A Math Sci. 87 (2011), 1–4.
  • M. Nakai, An application of capacitary functions to an inverse inclusion problem, Hiroshima Math. J. 41 (2011), 223–233.
  • M. Nakai and S. Segawa, Types of afforested surfaces, Kodai Math. J. 32 (2009), 109–116.
  • M. Parreau, Sur les moyennes des fonctions harmoniques et analytiques et la classification des surfaces de Riemann, Ann. Inst. Fourier 3 (1951/1952), 103–197.
  • L. Sario and M. Nakai, Classification theory of Riemann surfaces, Die Grundlehren der mathematischen Wissenschaften, Band 164, Springer-Verlag, New York-Berlin, 1970.