Atomic decomposition of real-variable type for Bergman spaces in the unit ball of $\mathbb{C}^n$

TL;DR

This paper establishes an atomic decomposition for Bergman spaces in the unit ball of complex n-dimensional space, providing a constructive proof based on estimates of Bergman metrics and kernels.

Contribution

It introduces a real-variable atomic decomposition for weighted Bergman spaces in the unit ball for all p between 0 and 1, with a constructive proof.

Findings

Atomic decomposition exists for all 0 < p ≤ 1 in Bergman spaces

Decomposition uses real-variable atoms and Bergman projection

Constructive proof based on sharp estimates of Bergman kernel and metric

Abstract

In this paper, we show that every (weighted) Bergman space $\mathcal{A}^p_{\alpha} (\mathbb{B}_n)$ in the complex ball admits an atomic decomposition of real-variable type for any $0 < p \le 1$ and $\alpha > -1.$ More precisely, for each $f \in \mathcal{A}^p_{\alpha} (\mathbb{B}_n)$ there exist a sequence of real-variable $(p, \8)_{\alpha}$-atoms $a_k$ and a scalar sequence $\{\lambda_k \}$ with $\sum_k | \lambda_k |^p < \8$ such that $f = \sum_k \lambda_k P_{\alpha} (a_k),$ where $P_{\alpha}$ is the Bergman projection from $L^2_{\alpha} (\mathbb{B}_n)$ onto $\mathcal{A}^2_{\alpha} (\mathbb{B}_n).$ The proof is constructive, and our construction is based on some sharp estimates about Bergman metric and Bergman kernel functions in $\mathbb{B}_n.$