Tohoku Mathematical Journal
2012
June
SECOND SERIES VOL. 64, NO. 2

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

Title SQUARE MEANS VERSUS DIRICHLET INTEGRALS FOR HARMONIC FUNCTIONS ON RIEMANN SURFACES

Author Hiroaki Masaoka and Mitsuru Nakai

(Received May 26, 2010, revised May 9, 2011)
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.

2000 Mathematics Subject Classification. Primary 30F20; Secondary 30F25, 30F15, 31A15.
Key words and phrases. afforested surface, Dirichlet finite, Hardy space, Joukowski coordinate, mean bounded, Parreau decomposition, quasibounded, singular, Wiener harmonic boundary.

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