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MÖBIUS INVARIANT FUNCTION SPACES AND DIRICHLET SPACES WITH SUPERHARMONIC WEIGHTS

Published online by Cambridge University Press:  12 July 2018

GUANLONG BAO
Affiliation:
Department of Mathematics, Shantou University, Shantou, Guangdong 515063, China email glbao@stu.edu.cn
JAVAD MASHREGHI
Affiliation:
Département de Mathématiques et de Statistique, Université Laval, 1045 avenue de la Médecine, Québec, QC G1V 0A6, Canada email javad.mashreghi@mat.ulaval.ca
STAMATIS POULIASIS
Affiliation:
Department of Mathematics, Aristotle University of Thessaloniki, 54124 Thessaloniki, Greece
HASI WULAN
Affiliation:
Department of Mathematics, Shantou University, Shantou, Guangdong 515063, China email wulan@stu.edu.cn

Abstract

Let ${\mathcal{D}}_{\unicode[STIX]{x1D707}}$ be Dirichlet spaces with superharmonic weights induced by positive Borel measures $\unicode[STIX]{x1D707}$ on the open unit disk. Denote by $M({\mathcal{D}}_{\unicode[STIX]{x1D707}})$ Möbius invariant function spaces generated by ${\mathcal{D}}_{\unicode[STIX]{x1D707}}$. In this paper, we investigate the relation among ${\mathcal{D}}_{\unicode[STIX]{x1D707}}$, $M({\mathcal{D}}_{\unicode[STIX]{x1D707}})$ and some Möbius invariant function spaces, such as the space $BMOA$ of analytic functions on the open unit disk with boundary values of bounded mean oscillation and the Dirichlet space. Applying the relation between $BMOA$ and $M({\mathcal{D}}_{\unicode[STIX]{x1D707}})$, under the assumption that the weight function $K$ is concave, we characterize the function $K$ such that ${\mathcal{Q}}_{K}=BMOA$. We also describe inner functions in $M({\mathcal{D}}_{\unicode[STIX]{x1D707}})$ spaces.

Type
Research Article
Copyright
© 2018 Australian Mathematical Publishing Association Inc. 

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Footnotes

G. Bao and H. Wulan were supported by the NNSF of China (No. 11720101003). G. Bao was also supported by the STU Scientific Research Foundation for Talents (No. NTF17020).

Current address: Department of Mathematics and Statistics, Texas Tech University, Lubbock, Texas 79409, USA stamatis.pouliasis@ttu.edu

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MÖBIUS INVARIANT FUNCTION SPACES AND DIRICHLET SPACES WITH SUPERHARMONIC WEIGHTS
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