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

Abstract

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 $𝐻^{𝑝}_{𝑤, 𝐿_1, 𝐿_2}(𝑥_1 × 𝑥_2)$ 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 $𝐻^{𝑝}_{𝑤, 𝐿_1, 𝐿_2}(𝑥_1 × 𝑥_2)$ introduced via area functions can be equivalently characterized by the Littlewood–Paley 𝑔-functions and $𝑔^{\ast}_{𝜆_1, 𝜆_2}$-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.