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January, 2019 Equivalence of Littlewood–Paley square function and area function characterizations of weighted product Hardy spaces associated to operators
Xuan Thinh DUONG, Guorong HU, Ji LI
J. Math. Soc. Japan 71(1): 91-115 (January, 2019). DOI: 10.2969/jmsj/78287828

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.

Funding Statement

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

Citation

Download Citation

Xuan Thinh DUONG. Guorong HU. Ji LI. "Equivalence of Littlewood–Paley square function and area function characterizations of weighted product Hardy spaces associated to operators." J. Math. Soc. Japan 71 (1) 91 - 115, January, 2019. https://doi.org/10.2969/jmsj/78287828

Information

Received: 16 June 2017; Published: January, 2019
First available in Project Euclid: 24 October 2018

zbMATH: 07056559
MathSciNet: MR3909916
Digital Object Identifier: 10.2969/jmsj/78287828

Subjects:
Primary: 42B25
Secondary: 42B30, 47B25

Rights: Copyright © 2019 Mathematical Society of Japan

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Vol.71 • No. 1 • January, 2019
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