Equivalence of Littlewood-Paley square function and area function characterizations of weighted product Hardy spaces associated to operators. (English) Zbl 1414.42022

Duong, Xuan Thinh; Hu, Guorong; Li, Ji

J. Math. Soc. Japan 71, No. 1, 91-115 (2019).

Summary: Let \(L_1\) and \(L_2\) be nonnegative self-adjoint operators acting on \(L^2(X_1)\) and \(L^2(X_2)\), respectively, where \(X_1\) and \(X_2\) are spaces of homogeneous type. 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 \(A_{\infty}(X_{1} \times X_{2})\). Our main result is that the spaces \(H^{p}_{w,L_{1},L_{2}}(X_{1}\times X_{2})\) introduced via area functions can be equivalently characterized by the Littlewood-Paley \(g\)-functions and \(g^{\ast}_{\lambda_{1}, \lambda_{2}}\)-functions, as well as the Peetre type maximal functions, without any further assumption beyond the Gaussian upper bounds on the heat kernels of \(L_1\) and \(L_2\). Our results are new even in the unweighted product setting.

MSC:

42B25	Maximal functions, Littlewood-Paley theory
42B30	\(H^p\)-spaces
47B25	Linear symmetric and selfadjoint operators (unbounded)

Keywords:

product Hardy spaces; non-negative self-adjoint operator; heat semigroup; Littlewood-Paley functions; space of homogeneous type