Composition Operators on Hardy–Orlicz Spaces on the Ball

Abstract

Abstract. We give embedding theorems for Hardy-Orlicz spaces on the ball and then apply our results to the study of the boundedness and the compactness of composition operators in this context. As one of the motivations of this work, we show that there exist some Hardy-Orlicz spaces, different from H ∞ , on which every composition operator is bounded.

Compact composition operators on the Hardy--Orlicz and weighted Bergman-Orlicz spaces on the ball

“…All the results of the present paper are based on characterizations of the boundedness and compactness of composition operators on Hardy-Orlicz and Bergman-Orlicz spaces ( [2,3]). As I already said, these characterizations essentially depend on the manner in which the Orlicz function grows.…”

“…The same problem in the Bergman-Orlicz case has not yet been completely solved. In several variables, the situation is much more surprizing, as we show in [2,3] that there exist some Hardy-Orlicz and Bergman-Orlicz spaces, "close" enough to H ∞ , on which every composition operator is bounded.…”

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

“…The next theorem, stated for ψ in the ∆ 2 -class, is a particular case of the one obtained in [11,14] for a larger class of Orlicz functions (see also [25,28] for the unit disc).…”

“…The following result specifies the topological and dual properties of Hardy-Orlicz and Bergman-Orlicz spaces, pointing out that if we intend to generalize the classical Hardy or Bergman spaces and provide with a refined scale of spaces up to H ∞ , then we must consider Banach spaces with less nice properties. 11,14]). Let α ≥ −1 and let ψ be an Orlicz function.…”

“…Remark. (1) Using Carleson measure theorems, it was already stated in [11,14] that every composition operator is bounded on A ψ α (B N ) and H ψ (B N ) whenever ψ satisfies the ∆ 2 -condition. Yet, whatever the Orlicz function, no explicit estimate of the norm of a composition operator where known for N > 1.…”