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Actions of the Möbius group on analytic functions

Volume 260 / 2021

Guanlong Bao, Kehe Zhu Studia Mathematica 260 (2021), 207-228 MSC: 30H20, 30H25, 30H30, 30H35, 30B10, 46E15. DOI: 10.4064/sm200924-21-12 Published online: 4 March 2021


In his PhD dissertation (1996), Ruhan Zhao introduced a new notion of weighted actions by the Möbius group on analytic functions on the unit disk indexed by a positive parameter $\alpha $ and proved that the so-called $\alpha $-Bloch space is maximal among all $\alpha $-Möbius invariant function spaces. In this paper we continue the study of $\alpha $-Möbius invariant function spaces. In particular, we identify the minimal non-trivial $\alpha $-Möbius invariant function space and prove the existence and uniqueness of a non-trivial $\alpha $-Möbius invariant semi-Hilbert space of analytic functions on the unit disk, thus answering two questions left open by Zhao.


  • Guanlong BaoDepartment of Mathematics
    Shantou University
    Shantou, Guangdong 515063, China
  • Kehe ZhuDepartment of Mathematics and Statistics
    State University of New York
    Albany, NY 12222, U.S.A.

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