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MÖBIUS INVARIANT FUNCTION SPACES AND DIRICHLET SPACES WITH SUPERHARMONIC WEIGHTS

  • GUANLONG BAO (a1), JAVAD MASHREGHI (a2), STAMATIS POULIASIS (a3) and HASI WULAN (a4)

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.

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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

  • GUANLONG BAO (a1), JAVAD MASHREGHI (a2), STAMATIS POULIASIS (a3) and HASI WULAN (a4)

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