Weighted product Hardy space associated with operators

Assuming that the operators L₁, L₂ are self-adjoint and {\mathrm{e}}^{?{{\displaystyle tL}}_i}(i=\mathrm{1},\mathrm{2}) satisfy the generalized Davies-Gaffney estimates, we shall prove that the weighted Hardy space {{\displaystyle H}}_{{{\displaystyle L}}_{\mathrm{1}},{{\displaystyle L}}_{\mathrm{2}},\omega }^{\mathrm{1}}({?}^{{{\displaystyle n}}_{\mathrm{1}}}\times {?}^{{{\displaystyle n}}_{\mathrm{2}}}) associated to operators L₁, L₂ on product domain, which is defined in terms of area function, has an atomic decomposition for some weight \omega.