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

2016/10/21 by Charpentier, Stéphane
#Complex Variables (math.CV) #FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.1610.06775

Abstract

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 Δ²--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 Δ²--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.

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