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

Abstract

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 $(p, \8)_α$-atoms $a_k$ and a scalar sequence $\{{\lambda }_k\}$ with $\sum_k | λ_k |^p < \8$ 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 ${\mathcal{A}}_{\alpha }^2({\mathbb{B}}_n).$ The proof is constructive, and our construction is based on some sharp estimates about Bergman metric and Bergman kernel functions in ${\mathbb{B}}_n.$

28 pages