Equivalence of Littlewood–Paley square function and area function characterizations of weighted product Hardy spaces associated to operators

Let L₁ and L₂ be nonnegative self-adjoint operators acting on L² (X₁ ) and L² (X₂ ), respectively, where X₁ and X₂ are spaces of homogeneous type. Assume that L₁ and L₂ have Gaussian heat kernel bounds. This paper aims to study some equivalent characterizations of the weighted product Hardy spaces H^p_w,L1,L2 (X₁ × X₂) associated to L₁ and L₂ , for p ∈ (0, ∞) and the weight w belongs to the product Muckenhoupt class A_∞(X₁ × X₂). Our main result is that the spaces H^p_w,L1,_L2 (X₁ × X₂) introduced via area functions can be equivalently characterized by the Littlewood–Paley g-functions and g^∗_λ1,_λ2 -functions, as well as the Peetre type maximal functions, without any further assumption beyond the Gaussian upper bounds on the heat kernels of L₁ and L₂ . Our results are new even in the unweighted product setting.