Weighted product Hardy space associated with operators
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Weighted product Hardy space associated with operators Qingquan DENG1 ,
Djalal Eddine GUEDJIBA1,2
1 School of Mathematics and Statistics, Hubei Province Key Laboratory of Mathematical Physics, Central China Normal University, Wuhan 430079, China 2 Department of Mathematics, University of Batna 2, 53 Route de Constantine, F´esdis, Batna 05078, Algeria
c Higher Education Press 2020
Abstract Assuming that the operators L1 , L2 are self-adjoint and e−tLi (i = 1, 2) satisfy the generalized Davies-Gaffney estimates, we shall prove that the weighted Hardy space HL1 1 ,L2 ,ω (Rn1 × Rn2 ) associated to operators L1 , L2 on product domain, which is defined in terms of area function, has an atomic decomposition for some weight ω. Keywords Produce Hardy space, Ap weights, Davies-Gaffney estimates MSC 42B20, 42B25, 46B70 1
Introduction
The classic theory of Hardy spaces on product domains was introduced by Gundy and Stein [20]. It is then extensively and deeply studied in many other important works, including the atomic decomposition, Calder´ on-Zygmund theory, and other related topics on such spaces, see, for example, [3–6,8,14– 18,21–24,27,34,39]. Moreover, the weighted theory of Hardy spaces on product domains was established by Fefferman [17], Krug [28], and Sato [35], one can see also [2,12,38,40] for related results. Recently, the study of Hardy spaces associated to operators on product domains was introduced and developed by many authors. Deng et al. [10] defined the product Hardy space HL1 (R × R) in terms of area integrals. Under the assumption that L generates an analytic semigroup and has a bounded holomorphic functional calculus on L2 (R), as well as the kernel of e−tL satisfies point-wise Poisson type upper bound, the authors proved that the Hardy space HL1 (R × R) has molecular decomposition by using the atomic decomposition of tent spaces on product domain. Moreover, the dual space of HL1 ∗ (R × R) Received October 24, 2019; accepted June 25, 2020 Corresponding author: Qingquan DENG, E-mail: [email protected]
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Qingquan DENG, Djalal Eddine GUEDJIBA
is BMOL (R × R), the BMO space associated to the operator L on product domains. The atomic decomposition for Hardy spaces associated to operators on product domain HL1 1 ,L2 (Rn1 × Rn2 ) was obtained by Duong et al. [13], where the Hardy space HL1 1 ,L2 (Rn1 × Rn2 ) is defined via area integral and they assumed that the heat kernels for nonnegative self-adjoint operators L1 and L2 satisfy point-wise Gaussian upper bounds. Later, in the work of Song and Tan [36], the authors assumed that L is the Schr¨odinger operator −∆ + V with nonnegative potential V, defined the Hardy spaces on product domain in terms of area integral, maximal functions, and atoms, and proved all these spaces are equivalent. We have to mention that in [36], the kernel of semigroup e−tL satisfies the point-wise Gaussian upper bound. Notice that the above Hardy spaces associated to operators are considered under the assumption that the heat kernel of the operators have suitable pointwi
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