Products of Functions in Hardy and Lipschitz or BMO Spaces

Bonami, Aline; Feuto, Justin

doi:10.1007/978-0-8176-4588-5_4

Cited by 15 publications

(32 citation statements)

References 14 publications

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“…where C is a positive constant, independent of F and G. This result essentially extends the corresponding ones in [3,5]; see also [8,4] for more related results on the div-curl lemma.…”

Section: Introductionsupporting

confidence: 79%

“…However, it should be pointed out that the decomposition obtained in (1.2) through (1.3) is not bilinear. Also, as was pointed out in [3], the range space on the right hand side of the decomposition in (1.3) is not sharp.…”

Section: Introductionmentioning

confidence: 83%

“…(ii) the associated kernel function K of T satisfies the following size and the following regularity conditions: there exists a positive constant C such that, for all (x, y, z) ∈ (R n ) 3 \ Ω,…”

Section: Now We Show That the Four Bilinear Operatorsmentioning

confidence: 99%

“…For the case p less than 1, Bonami and Feuto [3] established a series of decompositions for the products of the Hardy spaces H p (R n ) with p ∈ (0, 1) and their dual spaces as follows:…”

Section: Introductionmentioning

confidence: 99%

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Bilinear decompositions of products of local Hardy and Lipschitz or BMO spaces through wavelets

Cao

Yang

2018

Commun. Contemp. Math.

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The aim of this article is to give a complete solution to the problem of the bilinear decompositions of the products of some Hardy spaces H p (R n ) and their duals in the case when p < 1 and near to 1, via wavelets, paraproducts and the theory of bilinear Calderón-Zygmund operators. Precisely, the authors establish the bilinear decompositions of the product spaces H p (R n ) ×Λ α (R n ) and H p (R n ) × Λ α (R n ), where, for all p ∈ ( n n+1 , 1) and α := n( 1 p −1), H p (R n ) denotes the classical real Hardy space, anḋ Λ α and Λ α denote the homogeneous, respectively, the inhomogeneous Lipschitz spaces. Sharpness of these two bilinear decompositions are also proved. As an application, the authors establish some div-curl lemmas at the endpoint case.

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“…where C is a positive constant, independent of F and G. This result essentially extends the corresponding ones in [3,5]; see also [8,4] for more related results on the div-curl lemma.…”

Section: Introductionsupporting

confidence: 79%

Section: Introductionmentioning

confidence: 83%

Section: Now We Show That the Four Bilinear Operatorsmentioning

confidence: 99%

“…For the case p less than 1, Bonami and Feuto [3] established a series of decompositions for the products of the Hardy spaces H p (R n ) with p ∈ (0, 1) and their dual spaces as follows:…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Bilinear decompositions of products of local Hardy and Lipschitz or BMO spaces through wavelets

Cao

Yang

2018

Commun. Contemp. Math.

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“…On another hand, observe that (E p Φ ) t (R n ) when p = t = 1 goes back to the amalgam space (L Φ , ℓ 1 )(R n ) introduced by Bonami and Feuto [11], where Φ(t) := t log(e+t) for any t ∈ [0, ∞), and the Hardy space H Φ * (R n ) associated with the amalgam space (L Φ , ℓ 1 )(R n ) was applied by Bonami and Feuto [11] to study the linear decomposition of the product of the Hardy space H 1 (R n ) and its dual space BMO (R n ). Another main motivation to introduce (HE q Φ ) t (R n ) in [81] exists in that it is a natural generalization of H Φ * (R n ) in [11]. In the last part of this section, we focus on the weak Orlicz-slice Hardy space (W HE q Φ ) t (R n ) built on the Orlicz-slice space (E q Φ ) t (R n ).…”

Section: Orlicz-slice Spacesmentioning

confidence: 97%

Weak Hardy-Type Spaces Associated with Ball Quasi-Banach Function Spaces II: Littlewood–Paley Characterizations and Real Interpolation

et al. 2019

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Let X be a ball quasi-Banach function space on R n . In this article, assuming that the powered Hardy-Littlewood maximal operator satisfies some Fefferman-Stein vector-valued maximal inequality on X as well as it is bounded on both the weak ball quasi-Banach function space WX and the associated space, the authors establish various Littlewood-Paley function characterizations of WH X (R n ) under some weak assumptions on the Littlewood-Paley functions. The authors also prove that the real interpolation intermediate space (H X (R n ), L ∞ (R n )) θ,∞ , between the Hardy space associated with X, H X (R n ), and the Lebesgue space L ∞ (R n ), is WH X 1/(1−θ) (R n ), where θ ∈ (0, 1). All these results are of wide applications. Particularly, when X := M p q (R n ) (the Morrey space), X := L p (R n ) (the mixed-norm Lebesgue space) and X := (E q Φ ) t (R n ) (the Orlicz-slice space), all these results are even new; when X := L Φ ω (R n ) (the weighted Orlicz space), the result on the real interpolation is new and, when X := L p(·) (R n ) (the variable Lebesgue space) and X := L Φ ω (R n ), the Littlewood-Paley function characterizations of WH X (R n ) obtained in this article improves the existing results via weakening the assumptions on the Littlewood-Paley functions.

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