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

Abstract

In this paper we show that, for any $0 < p \le 1$ and $\alpha > -1$, every (weighted) Bergman space $\mathcal{A}^p_{\alpha} (\mathbb{B}_n)$ admits an atomic decomposition of real-variable type. More precisely, for each $f \in \mathcal{A}^p_{\alpha} (\mathbb{B}_n)$ there exist a sequence of $(p, \infty)_{\alpha}$-atoms $a_k$ with compact support and a scalar sequence $\{\lambda_k \}$ such that $f = \sum_k \lambda_k a_k$ in the sense of distribution and $\sum_k | \lambda_k |^p \lesssim \| f \|^p_{p, \alpha};$ and moreover, $f = \sum_k \lambda_k P_{\alpha} ( a_k)$ in $\mathcal{A}^p_{\alpha} (\mathbb{B}_n),$ where $P_{\alpha}$ is the orthogonal 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 analysis inside the unit ball $\mathbb{B}_n$ associated with a quasimetric.