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