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

2017/06/19 by Duong, Xuan Thinh, Hu, Guorong, Li, Ji
#Classical Analysis and ODEs (math.CA) #FOS: Mathematics

paper · doi:10.48550/arxiv.1706.05803

Abstract

Let L₁ and L₂ be non-negative 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 Littlewood-Paley g-functions, Littlewood-Paley g^∗_λ₁, λ₂-functions, and Peetre type maximal functions, without any further assumptions beyond the Gaussian upper bounds on the heat kernels of L₁ and L₂. Our results are new even in the unweighted product setting.

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