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

Research output: Contribution to journalArticleResearchpeer-review

Abstract

Let L1 and L2 be nonnegative self-adjoint operators acting on L2 (X1 ) and L2 (X2 ), respectively, where X1 and X2 are spaces of homogeneous type. Assume that L1 and L2 have Gaussian heat kernel bounds. This paper aims to study some equivalent characterizations of the weighted product Hardy spaces Hpw,L1,L2 (X1 × X2) associated to L1 and L2 , for p ∈ (0, ∞) and the weight w belongs to the product Muckenhoupt class A(X1 × X2). Our main result is that the spaces Hpw,L1,L2 (X1 × X2) introduced via area functions can be equivalently characterized by the Littlewood–Paley g-functions and gλ1,λ2 -functions, as well as the Peetre type maximal functions, without any further assumption beyond the Gaussian upper bounds on the heat kernels of L1 and L2 . Our results are new even in the unweighted product setting.

LanguageEnglish
Pages91-115
Number of pages25
JournalJournal of the Mathematical Society of Japan
Volume71
Issue number1
DOIs
Publication statusPublished - 1 Jan 2019

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Littlewood-Paley Function
Square Functions
Product Space
Heat Kernel
Hardy Space
Equivalence
Space of Homogeneous Type
Gaussian Kernel
Maximal Function
G-function
Operator
Self-adjoint Operator
Non-negative
Upper bound

Cite this

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abstract = "Let L1 and L2 be nonnegative self-adjoint operators acting on L2 (X1 ) and L2 (X2 ), respectively, where X1 and X2 are spaces of homogeneous type. Assume that L1 and L2 have Gaussian heat kernel bounds. This paper aims to study some equivalent characterizations of the weighted product Hardy spaces Hpw,L1,L2 (X1 × X2) associated to L1 and L2 , for p ∈ (0, ∞) and the weight w belongs to the product Muckenhoupt class A∞(X1 × X2). Our main result is that the spaces Hpw,L1,L2 (X1 × X2) introduced via area functions can be equivalently characterized by the Littlewood–Paley g-functions and g∗λ1,λ2 -functions, as well as the Peetre type maximal functions, without any further assumption beyond the Gaussian upper bounds on the heat kernels of L1 and L2 . Our results are new even in the unweighted product setting.",
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Equivalence of Littlewood–Paley square function and area function characterizations of weighted product Hardy spaces associated to operators. / Duong, Xuan Thinh; Hu, Guorong; Li, Ji.

In: Journal of the Mathematical Society of Japan, Vol. 71, No. 1, 01.01.2019, p. 91-115.

Research output: Contribution to journalArticleResearchpeer-review

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AU - Hu, Guorong

AU - Li, Ji

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N2 - Let L1 and L2 be nonnegative self-adjoint operators acting on L2 (X1 ) and L2 (X2 ), respectively, where X1 and X2 are spaces of homogeneous type. Assume that L1 and L2 have Gaussian heat kernel bounds. This paper aims to study some equivalent characterizations of the weighted product Hardy spaces Hpw,L1,L2 (X1 × X2) associated to L1 and L2 , for p ∈ (0, ∞) and the weight w belongs to the product Muckenhoupt class A∞(X1 × X2). Our main result is that the spaces Hpw,L1,L2 (X1 × X2) introduced via area functions can be equivalently characterized by the Littlewood–Paley g-functions and g∗λ1,λ2 -functions, as well as the Peetre type maximal functions, without any further assumption beyond the Gaussian upper bounds on the heat kernels of L1 and L2 . Our results are new even in the unweighted product setting.

AB - Let L1 and L2 be nonnegative self-adjoint operators acting on L2 (X1 ) and L2 (X2 ), respectively, where X1 and X2 are spaces of homogeneous type. Assume that L1 and L2 have Gaussian heat kernel bounds. This paper aims to study some equivalent characterizations of the weighted product Hardy spaces Hpw,L1,L2 (X1 × X2) associated to L1 and L2 , for p ∈ (0, ∞) and the weight w belongs to the product Muckenhoupt class A∞(X1 × X2). Our main result is that the spaces Hpw,L1,L2 (X1 × X2) introduced via area functions can be equivalently characterized by the Littlewood–Paley g-functions and g∗λ1,λ2 -functions, as well as the Peetre type maximal functions, without any further assumption beyond the Gaussian upper bounds on the heat kernels of L1 and L2 . Our results are new even in the unweighted product setting.

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