Small Bergman-Orlicz and Hardy-Orlicz spaces, and their composition operators

We show that the weighted Bergman-Orlicz space A α ψ coincides with some weighted Banach space of holomorphic functions if and only if the Orlicz function ψ satisfies the so-called Δ 2 -condition. In addition we prove that this condition characterizes those A α ψ on which every composition operator...

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Bibliographic Details
Published in:Mathematische Zeitschrift 2019-12, Vol.293 (3-4), p.1287-1314
Main Author: Charpentier, S.
Format: Article
Language:English
Subjects:
Analytic functions
Banach spaces
Composition
Mathematics
Mathematics and Statistics
Operators (mathematics)
Orlicz space
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Summary:We show that the weighted Bergman-Orlicz space A α ψ coincides with some weighted Banach space of holomorphic functions if and only if the Orlicz function ψ satisfies the so-called Δ 2 -condition. In addition we prove that this condition characterizes those A α ψ on which every composition operator is bounded or order bounded into the Orlicz space L α ψ . This provides us with estimates of the norm and the essential norm of composition operators on such spaces. We also prove that when ψ satisfies the Δ 2 -condition, a composition operator is compact on A α ψ if and only if it is order bounded into the so-called Morse–Transue space M α ψ . Our results stand in the unit ball of C N .
ISSN:0025-5874
1432-1823
DOI:10.1007/s00209-019-02240-w