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

## TL;DR

This paper establishes that weighted product Hardy spaces associated with certain operators can be characterized equivalently by various square and maximal functions, broadening understanding of these spaces without extra assumptions.

## Contribution

It proves the equivalence of different characterizations of weighted product Hardy spaces linked to operators under Gaussian heat kernel bounds, even in unweighted cases.

## Key findings

- Equivalence of area function and Littlewood-Paley characterizations.
- Extension to weighted spaces with Muckenhoupt weights.
- Results hold without additional assumptions beyond heat kernel bounds.

## 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 $H^{p}_{w,L_{1},L_{2}}(X_{1}\times X_{2})$ 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 Littlewood-Paley $g$-functions, Littlewood-Paley $g^{\ast}_{\lambda_{1}, \lambda_{2}}$-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.