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

Xuan Thinh DUONG, Guorong HU, and Ji LI

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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.


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

Article information

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

Received: 16 June 2017
First available in Project Euclid: 24 October 2018

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 42B25: Maximal functions, Littlewood-Paley theory
Secondary: 42B30: $H^p$-spaces 47B25: Symmetric and selfadjoint operators (unbounded)

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


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.

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