Weighted product Hardy space associated with operators

Assuming that the operators L1, L2 are self-adjoint and e−tLi\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\rm{e}}^{ - t{L_i}}}$$\end{document} (i = 1, 2) satisfy the generalized Davies-Gaffney estimates, we shall prove that the weighted Hardy space HL1,L2,ω1(ℝn1×ℝn2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H_{{L_1},{L_2},\omega }^1({\mathbb{R}^{{n_1}}} \times {\mathbb{R}^{{n_2}}})$$\end{document} associated to operators L1, L2 on product domain, which is defined in terms of area function, has an atomic decomposition for some weight ω.