Qp spaces on Riemann surfaces

We study the function spaces Q(p)(R) defined on a Riemann surface R, which were earlier introduced in the unit disk of the complex plane. The nesting property Q(p)(R) subset of or equal to Q(q)(R) for 0 < p < q < infinity is shown in case of arbitrary hyperbolic Riemann surfaces. Further, it is proved that the classical Dirichlet space AD(R) subset of or equal to Q(p)(R) for any p, 0 < p < infinity, thus sharpening T. Metzger's well-known result AD(R) subset of or equal to BMOA(R). Also the first author's result AD(R) subset of or equal to VMOA(R) for a regular Riemann surface R is sharpened by showing that, in fact, AD(R) subset of or equal to Q(p,0)(R) for all p, 0 < p < infinity. The relationships between ep(R) and various generalizations of the Bloch space on R are considered. Finally we show that Q(p)(R) is a Banach space for 0 < p < infinity.