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2012 Square means versus Dirichlet integrals for harmonic functions on Riemann surfaces
Hiroaki Masaoka, Mitsuru Nakai
Tohoku Math. J. (2) 64(2): 233-259 (2012). DOI: 10.2748/tmj/1341249373


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.


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Hiroaki Masaoka. Mitsuru Nakai. "Square means versus Dirichlet integrals for harmonic functions on Riemann surfaces." Tohoku Math. J. (2) 64 (2) 233 - 259, 2012.


Published: 2012
First available in Project Euclid: 2 July 2012

zbMATH: 1262.30038
MathSciNet: MR2948821
Digital Object Identifier: 10.2748/tmj/1341249373

Primary: 30F20
Secondary: 30F15 , 30F25 , 31A15

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

Rights: Copyright © 2012 Tohoku University


Vol.64 • No. 2 • 2012
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