Maximal and area integral characterizations of Bergman spaces in the unit ball of $\mathbb{C}^n$

TL;DR

This paper provides new maximal and area integral characterizations of Bergman spaces in the unit ball of complex n-space, using derivatives and gradients, and extends these results to Besov spaces with atomic decompositions.

Contribution

It introduces novel maximal and area integral characterizations of Bergman and Besov spaces, including atomic decompositions with respect to Carleson tubes.

Findings

New maximal and area integral characterizations of Bergman spaces.

Characterizations involve radial, complex, and invariant gradients.

Atomic decomposition for Bergman spaces using Carleson tubes.

Abstract

In this paper, we present maximal and area integral characterizations of Bergman spaces in the unit ball of $\mathbb{C}^n.$ The characterizations are in terms of maximal functions and area integral functions on Bergman balls involving the radial derivative, the complex gradient, and the invariant gradient. As an application, we obtain new maximal and area integral characterizations of Besov spaces. Moreover, we give an atomic decomposition of real-variable type with respect to Carleson tubes for Bergman spaces.