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

  • S. CharpentierEmail author
Article
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Abstract

We show that the weighted Bergman-Orlicz space \(A_{\alpha }^{\psi }\) coincides with some weighted Banach space of holomorphic functions if and only if the Orlicz function \(\psi \) satisfies the so-called \(\Delta ^{2}\)-condition. In addition we prove that this condition characterizes those \(A_{\alpha }^{\psi }\) on which every composition operator is bounded or order bounded into the Orlicz space \(L_{\alpha }^{\psi }\). This provides us with estimates of the norm and the essential norm of composition operators on such spaces. We also prove that when \(\psi \) satisfies the \(\Delta ^{2}\)-condition, a composition operator is compact on \(A_{\alpha }^{\psi }\) if and only if it is order bounded into the so-called Morse–Transue space \(M_{\alpha }^{\psi }\). Our results stand in the unit ball of \({\mathbb {C}}^{N}\).

Keywords

Hardy-Orlicz space Several complex variables Bergman-Orlicz space Weighted Banach space of holomorphic functions Composition operator Several complex variables 

Mathematics Subject Classification

Primary 47B33 Secondary 30H05 32C22 46E15 

Notes

Acknowledgements

The author is grateful to the referee whose suggestions have improved both the manuscript and its presentation.

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Institut de Mathématiques, UMR 7373Aix-Marseille UniversitéMarseille Cedex 13France

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