Weak Hardy-type spaces associated with ball quasi-Banach function spaces I: Decompositions with applications to boundedness of Calderón-Zygmund operators

Zhang, Yangyang; Yang, Dachun; Yuan, Wen; Wang, Songbai

doi:10.1007/s11425-019-1645-1

Cited by 64 publications

(48 citation statements)

References 75 publications

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“…where m N ( f ) is as in (4.27) and N ∈ N. From this, [53, Section 7.6 and Theorem 5.3] (see also [58,63] for some corrections), it follows that, if N is sufficiently large, then, for any given ϕ ∈ S(R n ),…”

Section: Bilinear Decomposition Of Hmentioning

confidence: 91%

Bilinear Decomposition and Divergence-Curl Estimates on Products Related to Local Hardy Spaces and Their Dual Spaces

Zhang,

Yang,

Yuan

2020

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Let p ∈ (0, 1), α := 1/p − 1 and, for any τ ∈ [0, ∞), Φ p (τ) := τ/(1 + τ 1−p ). Let H p (R n ), h p (R n ) and Λ nα (R n ) be, respectively, the Hardy space, the local Hardy space and the inhomogeneous Lipschitz space on R n . In this article, applying the inhomogeneous renormalization of wavelets, the authors establish a bilinear decomposition for multiplications of elements in h p (R n ) [or H p (R n )] and Λ nα (R n ), and prove that these bilinear decompositions are sharp in some sense. As applications, the authors also obtain some estimates of the product of elements in the local Hardy space h p (R n ) with p ∈ (0, 1] and its dual space, respectively, with zero ⌊nα⌋-inhomogeneous curl and zero divergence, where ⌊nα⌋ denotes the largest integer not greater than nα. Moreover, the authors find new structures of h Φ p (R n ) and) with equivalent quasi-norms, and also prove that the dual spaces of both h Φ p (R n ) and h p (R n ) coincide. These results give a complete picture on the multiplication between the local Hardy space and its dual space.

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Section: Bilinear Decomposition Of Hmentioning

confidence: 91%

Bilinear Decomposition and Divergence-Curl Estimates on Products Related to Local Hardy Spaces and Their Dual Spaces

Zhang,

Yang,

Yuan

2020

Preprint

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“…From the definition of L r (R n ), we easily deduce that L r (R n ), where r ∈ (0, ∞) n , is a ball quasi-Banach function space. But, L r (R n ) may not be a quasi-Banach function space (see, for instance, [79,Remark 7.20]). For any given r := (r 1 , .…”

Section: Mixed-norm Lebesgue Spacesmentioning

confidence: 99%

“…They are less restrictive than the classical Banach function spaces introduced in the book [7, Chapter 1]. For more studies on ball quasi-Banach function spaces, we refer the reader to [65,64,68,77,78,17] for the Hardy space associated with ball quasi-Banach function spaces, to [79,33,75] for the boundedness of operators on ball quasi-Banach function spaces, and to [41,42,76,37,72] for the applications of ball quasi-Banach function spaces.…”

Section: Introductionmentioning

confidence: 99%

Generalization in Ball Banach Function Spaces of Brezis--Van Schaftingen--Yung Formulae with Applications to Fractional Sobolev and Gagliardo--Nirenberg Inequalities

Ding¹,

Lin²,

Yang³

et al. 2021

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Let X be a ball Banach function space on R n . In this article, under the mild assumption that the Hardy-Littlewood maximal operator is bounded on the associated space X ′ of X, the authors prove that, for anywith the positive equivalence constants independent of f , where q ∈ (0, ∞) is an index depending on the space X, and |E| denotes the Lebesgue measure of a measurable set E ⊂ R n . Particularly, when ), the above estimate holds true for any given q ∈ [1, p], which when q = p is exactly the recent surprising formula of H. Brezis, J. Van Schaftingen, and P.-L. Yung, and which even when q < p is new. This generalization has a wide range of applications and, particularly, enables the authors to establish new fractional Sobolev and Gagliardo-Nirenberg inequalities in various function spaces, including Morrey spaces, mixed-norm Lebesgue spaces, variable Lebesgue spaces, weighted Lebesgue spaces, Orlicz spaces, and Orlicz-slice (generalized amalgam) spaces, and, even in all these special cases, the obtained results are new. The proofs of these results strongly depend on the Poincaré inequality, the extrapolation, the exact operator norm on X ′ of the Hardy-Littlewood maximal operator, and the geometry of R n .

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“…The following extrapolation theorem is just [78,Lemma 7.34], which is a slight variant of a special case of [25,Theorem 4.6] via replacing Banach function spaces by ball Banach function spaces.…”

Section: Sufficiency and Necessity Of Boundedness Of Commutatorsmentioning

confidence: 99%

Compactness Characterizations of Commutators on Ball Banach Function Spaces

Tao¹,

Yang²,

Yuan³

et al. 2021

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Let X be a ball Banach function space on R n . Let Ω be a Lipschitz function on the unit sphere of R n , which is homogeneous of degree zero and has mean value zero, and let T Ω be the convolutional singular integral operator with kernel Ω(•)/| • | n . In this article, under the assumption that the Hardy-Littlewood maximal operator M is bounded on both X and its associated space, the authors prove that the commutator [b, T Ω ] is compact on X if and only if b ∈ CMO(R n ). To achieve this, the authors mainly employ three key tools: some elaborate estimates, given in this article, on the norm in X of the commutators and the characteristic functions of some measurable subset, which are implied by the assumed boundedness of M on X and its associated space as well as the geometry of R n ; the complete John-Nirenberg inequality in X obtained by Y. Sawano et al.; the generalized Fréchet-Kolmogorov theorem on X also established in this article. All these results have a wide range of applications. Particularly, even when X := L p(•) (R n ) (the variable Lebesgue space), X := L p (R n ) (the mixed-norm Lebesgue space), X := L Φ (R n ) (the Orlicz space), and X := (E q Φ ) t (R n ) (the Orlicz-slice space or the generalized amalgam space), all these results are new.

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Weak Hardy-type spaces associated with ball quasi-Banach function spaces I: Decompositions with applications to boundedness of Calderón-Zygmund operators

Cited by 64 publications

References 75 publications

Bilinear Decomposition and Divergence-Curl Estimates on Products Related to Local Hardy Spaces and Their Dual Spaces

Bilinear Decomposition and Divergence-Curl Estimates on Products Related to Local Hardy Spaces and Their Dual Spaces

Generalization in Ball Banach Function Spaces of Brezis--Van Schaftingen--Yung Formulae with Applications to Fractional Sobolev and Gagliardo--Nirenberg Inequalities

Compactness Characterizations of Commutators on Ball Banach Function Spaces

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