## Journal of the Mathematical Society of Japan

### 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 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.

#### Note

The first and third authors were supported by DP 160100153. The second author is the corresponding author.

#### Article information

Source
J. Math. Soc. Japan, Volume 71, Number 1 (2019), 91-115.

Dates
First available in Project Euclid: 24 October 2018

https://projecteuclid.org/euclid.jmsj/1540368034

Digital Object Identifier
doi:10.2969/jmsj/78287828

Mathematical Reviews number (MathSciNet)
MR3909916

Zentralblatt MATH identifier
07056559

#### Citation

DUONG, Xuan Thinh; HU, Guorong; LI, Ji. Equivalence of Littlewood–Paley square function and area function characterizations of weighted product Hardy spaces associated to operators. J. Math. Soc. Japan 71 (2019), no. 1, 91--115. doi:10.2969/jmsj/78287828. https://projecteuclid.org/euclid.jmsj/1540368034

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