Atomic decomposition of real-variable type for Bergman spaces in the unit ball of $\mathbb{C}^n$ (1303.2182v1)

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 $\mathbb{B}_n.$