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

In this paper, we show that every (weighted) Bergman space ${\mathcal{A}}_{\alpha }^p({\mathbb{B}}_n)$ 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 $f\in {\mathcal{A}}_{\alpha }^p({\mathbb{B}}_n)$ there exist a sequence of real-variable $(p,\text{8}{)}_{\alpha }$-atoms $a_k$ and a scalar sequence $\{{\lambda }_k\}$ with ${\sum }_k\mathrm{\mid }{\lambda }_k{\mathrm{\mid }}^p<\text{8}$ such that $f = \sum_k \lambda_k P_{\alpha} (a_k),$where$P_{\alpha}$ is the Bergman projection from $L_{\alpha }^2({\mathbb{B}}_n)$ 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 ${\mathbb{B}}_n\mathrm{.}$