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Qp Spaces on Riemann Surfaces

Published online by Cambridge University Press:  20 November 2018

Rauno Aulaskari ,
Yuzan He ,
Juha Ristioja  and
Ruhan Zhao
Rauno Aulaskari
Affiliation:
Department of Mathematics University of JoensuuP.O. Box 111 FIN-80101 Joensuu Finland
Yuzan He
Affiliation:
Department of Mathematics University of JoensuuP.O. Box 111 FIN-80101 Joensuu Finland
Juha Ristioja
Affiliation:
Department of Mathematics University of JoensuuP.O. Box 111 FIN-80101 Joensuu Finland
Ruhan Zhao
Affiliation:
Institute of Mathematics AcademiaSinica Beijing 100080 People's Republic of China
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Abstract

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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 $.

Keywords

30D4530D5030F35
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 50 , Issue 3 , 01 June 1998 , pp. 449 - 464
Copyright
Copyright © Canadian Mathematical Society 1998

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