Atomic decomposition and Weak Factorization for Bergman-Orlicz spaces

TL;DR

This paper develops atomic decomposition and weak factorization theorems for Bergman-Orlicz spaces on the unit ball in complex space, generalizing classical results to more flexible growth functions.

Contribution

It introduces atomic decomposition for Bergman-Orlicz spaces with convex or concave growth functions and establishes new weak factorization theorems involving these spaces and the Bloch space.

Findings

Atomic decomposition for Bergman-Orlicz spaces established.

Weak factorization theorems involving Bloch and Bergman-Orlicz spaces proved.

Generalization of classical Bergman space results to Orlicz setting.

Abstract

For $\mathbb B^n$ the unit ball of $\mathbb C^n$, we consider Bergman-Orlicz spaces of holomorphic functions in $L^\Phi_\alpha(\mathbb B^n)$, which are generalizations of classical Bergman spaces. We obtain atomic decomposition for functions in the Bergman-Orlicz space $\mathcal A^\Phi_\alpha (\mathbb B^n)$ where $\Phi$ is either convex or concave growth function. We then prove weak factorization theorems involving the Bloch space and a Bergman-Orlicz space and also weak factorization theorems involving two Bergman-Orlicz spaces.