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

Abstract

Let $L_1$ and $L_2$ be non-negative 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 $H_{w,L_1,L_2}^p(X_1\times X_2)$ 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 $H_{w,L_1,L_2}^p(X_1\times X_2)$ introduced via area functions can be equivalently characterized by Littlewood-Paley $g$-functions, Littlewood-Paley $g_{{\lambda }_1,{\lambda }_2}^{\ast }$-functions, and Peetre type maximal functions, without any further assumptions 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.

18 pages