Tohoku Mathematical Journal

Square means versus Dirichlet integrals for harmonic functions on Riemann surfaces

Hiroaki Masaoka and Mitsuru Nakai

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Abstract

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

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

Dates
First available in Project Euclid: 2 July 2012

Permanent link to this document
https://projecteuclid.org/euclid.tmj/1341249373

Digital Object Identifier
doi:10.2748/tmj/1341249373

Mathematical Reviews number (MathSciNet)
MR2948821

Zentralblatt MATH identifier
1262.30038

Subjects
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]

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

Citation

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. https://projecteuclid.org/euclid.tmj/1341249373


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References

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