On new decomposition theorems in some analytic function spaces in bounded pseudoconvex domains

We provide new sharp decomposition theorems for multifunctional Bergman spaces in the unit ball and bounded pseudoconvex domains with smooth boundary expanding known results from the unit ball.
Namely we prove that $ \prod \limits_{j=1}^{m}||f_{j}|| _{X_{j}} \asymp ||f_{1} \dots f_{m}||_{A_{\alpha}^{p}}$ for various $(X_{j})$ spaces of analytic functions in bounded pseudoconvex domains with smooth boundary where $f, f_{j}, j=1,\dots, m$ are analytic functions and where $A_{\alpha}^{p}, 0 <p< \infty, \alpha>-1$ is a Bergman space. This in particular also extend in various directions a known theorem on atomic decomposition of Bergman $A^{p}_{\alpha}$ spaces.