Small Bergman-Orlicz and Hardy-Orlicz spaces, and their composition operators - Charpentier, S. (Mathematische Zeitschrift, 2019)
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 .