Qp Spaces on Riemann Surfaces

Aulaskari, Rauno ; He, Yuzan ; Ristioja, Juha ; Zhao, Ruhan

Canadian journal of mathematics, Tome 50 (1998) no. 3, pp. 449-464

Résumé

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 ${{Q}_{p}}(R)\,\subseteq \,{{Q}_{_{q}}}(R)$ for $0\,<\,p\,<\,q\,<\,\infty $ is shown in case of arbitrary hyperbolic Riemann surfaces. Further, it is proved that the classical Dirichlet space $\text{AD(}R\text{)}\,\subseteq \,{{Q}_{p}}(R)$ for any $p$ , $0\,<\,p\,<\,\infty $ , thus sharpening T. Metzger's well-known result $\text{AD(}R\text{)}\,\subseteq \,\text{BMOA}(R)$ . Also the first author's result $\text{AD}(R)\,\subseteq \,\text{VMOA}(R)$ for a regular Riemann surface $R$ is sharpened by showing that, in fact, $\text{AD(}R\text{)}\,\subseteq \,{{Q}_{p,0}}(R)$ for all $p$ , $0\,<\,p\,<\,\infty $ . The relationships between ${{Q}_{p}}(R)$ 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\,<\,\infty $ .

Détail

DOI : 10.4153/CJM-1998-024-4

Mots-clés : 30D45, 30D50, 30F35

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