2019 Volume 71 Issue 1 Pages 91-115
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.
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