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Assume that $L_{1}$ and $L_{2}$ have Gaussian heat kernel bounds. This paper aims to study some equivalent characterizations of the weighted product Hardy spaces $H^{p}_{w,L_{1},L_{2}}(X_{1}\\times X_{2})$ associated to $L_{1}$ and $L_{2}$, for $p \\in (0, \\infty)$ and the weight $w$ belongs to the product Muckenhoupt class $A_{\\infty}(X_{1} \\times X_{2})$. 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Assume that $L_{1}$ and $L_{2}$ have Gaussian heat kernel bounds. This paper aims to study some equivalent characterizations of the weighted product Hardy spaces $H^{p}_{w,L_{1},L_{2}}(X_{1}\\times X_{2})$ associated to $L_{1}$ and $L_{2}$, for $p \\in (0, \\infty)$ and the weight $w$ belongs to the product Muckenhoupt class $A_{\\infty}(X_{1} \\times X_{2})$. 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