Surfaces carrying sufficiently many Dirichlet finite harmonic functions that are automatically bounded. (English) Zbl 1264.30027

Let \(H(R)\) be the linear space of harmonic functions on a Riemann surface \(R\). Let \(HB(R)\) and \(HD(R)\) be the linear subspaces of \(H(R)\) consisting of the bounded harmonic functions on \(R\) and the harmonic functions on \(R\) with finite Dirichlet integral, respectively. In general, there is no inclusion relation between these two subspaces, but there are a lot of Riemann surfaces satisfying \(HB(R)\subset HD(R)\) with the linear dimension \(\dim HB(R)<\infty\) or \(HD(R)\subset HB(R)\) with \(\dim HD(R)<\infty\). In this paper the author studies the problem whether the relation \(HD(R)\subset HB(R)\) always implies the finiteness of \(\dim HD(R)\) or not, and shows the existence of a Riemann surface \(R\) such that \(HD(R)\subset HB(R)\) with \(\dim HD(R)=\infty\). For the construction of such a Riemann surface, the author makes use of the Sario-Tôki disc. With respect to the converse inclusion relation, it is already shown by the author that there exists a Riemann surface \(R\) such that \(HB(R)\subset HD(R)\) with \(\dim HB(R)=\infty\).


30F20 Classification theory of Riemann surfaces
30F25 Ideal boundary theory for Riemann surfaces
30F15 Harmonic functions on Riemann surfaces
31A05 Harmonic, subharmonic, superharmonic functions in two dimensions
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