Atomic decomposition and weak factorization in generalized Hardy spaces of closed forms

Bonami, Aline; Feuto, Justin; Grellier, Sandrine; Ky, Luong Dang

doi:10.1016/j.bulsci.2017.07.004

Cited by 10 publications

(4 citation statements)

References 19 publications

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“…Moreover, Bonami et al in [7,6] deduced that (1.1) is sharp in some sense and, in [8], they proved that, in dimension one, H log (R) is indeed the smallest space, in the sense of the inclusion of sets, having the above property. Recently, in [3,10], a bilinear decomposition theorem for multiplications of functions in H p (R n ) and its dual space C α (R n ) was established when p ∈ (0, 1) and α := 1/p−1, and the sharpness in some sense of this bilinear decomposition was also obtained therein.…”

Section: Introductionmentioning

confidence: 99%

Real-variable characterizations of Orlicz-slice Hardy spaces

et al. 2019

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In this article, the authors first introduce a class of Orlicz-slice spaces which generalize the slice spaces recently studied by P. Auscher et al. Based on these Orlicz-slice spaces, the authors introduce a new kind of Hardy type spaces, the Orlicz-slice Hardy spaces, via the radial maximal functions. This new scale of Orlicz-slice Hardy spaces contains the variant of the Orlicz-Hardy space of A. Bonami and J. Feuto as well as the Hardy-amalgam space of Z. V. de P. Ablé and J. Feuto as special cases. Their characterizations via the atom, the molecule, various maximal functions, the Poisson integral and the Littlewood-Paley functions are also obtained. As an application of these characterizations, the authors establish their finite atomic characterizations, which further induce a description of their dual spaces and a criterion on the boundedness of sublinear operators from these Orlicz-slice Hardy spaces into a quasi-Banach space. Then, applying this criterion, the authors obtain the boundedness of δ-type Calderón-Zygmund operators on these Orlicz-slice Hardy spaces. All these results are new even for slice Hardy spaces and, moreover, for Hardy-amalgam spaces, the Littlewood-Paley function characterizations, the dual spaces and the boundedness of δ-type Calderón-Zygmund operators on these Hardy-type spaces are also new. )(R n ), where Φ(t) := t log(e+t) for any t ∈ [0, ∞) is an Orlicz function, and applied these Hardy-type spaces to study the linear decomposition of the product of the Hardy space H 1 (R n ) and its dual space BMO (R n ) as well as the local Hardy space h 1 (R n ) and its dual space bmo (R n ). Moreover, very recently, Cao et al. [12] applied h Φ * (R n ) to study the bilinear decomposition of the product of the local Hardy space h 1 (R n ) and its dual space bmo (R n ). Recall that both the Hardy type spaces H Φ * (R n ) and h Φ * (R n ) were defined in [8] via the (local) radial maximal functions, while h Φ * (R n ) in [12] was defined via the local grand maximal function. Moreover, no other real-variable characterizations of both the Hardy type spaces H Φ * (R n ) and h Φ * (R n ) are known so far. On the other hand, recently, to study the classification of weak solutions in the natural classes for the boundary value problems of a t-independent elliptic system in the upper plane, Auscher and Mourgoglou [6] introduced the slice spaces E

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Section: Introductionmentioning

confidence: 99%

Real-variable characterizations of Orlicz-slice Hardy spaces

et al. 2019

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“…The question of the optimality of the result was raised in this article, to know whether H ϕ (R n ) can be replaced by a smaller vector space. The pointwise multiplier theorem of Nakai and Yabuta [31] allowed the authors in [7] to answer that the smallest Banach space containing H ϕ (R n ) is in some sense the smallest Banach space containing these products. Optimality was deduced in one dimension in [10] from an exact factorization and in [7] for n ≥ 2 from a weak factorization.…”

Section: Introductionmentioning

confidence: 99%

“…The pointwise multiplier theorem of Nakai and Yabuta [31] allowed the authors in [7] to answer that the smallest Banach space containing H ϕ (R n ) is in some sense the smallest Banach space containing these products. Optimality was deduced in one dimension in [10] from an exact factorization and in [7] for n ≥ 2 from a weak factorization. More related progress on this subject over R n can be found in [12,6,45,46].…”

Section: Introductionmentioning

confidence: 99%

Products and Commutators of Martingales in $H_1$ and ${\rm BMO}$

Bonami¹,

Jiao²,

Xie³

et al. 2023

Preprint

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Let f := ( f n ) n∈Z + and g := (g n ) n∈Z + be two martingales related to the probability space (Ω, F , P) equipped with the filtration (F n ) n∈Z + . Assume that f is in the martingale Hardy space H 1 and g is in its dual space, namely the martingale BMO. Then the semi-martingale f • g := ( f n g n ) n∈Z + may be written as the sumThe authors prove that L( f, g) is a process with bounded variation and limit in L 1 , while G( f, g) belongs to the martingale Hardy-Orlicz space H log associated with the Orlicz functionThe above bilinear decomposition L 1 + H log is sharp in the sense that, for particular martingales, the space L 1 + H log cannot be replaced by a smaller space having a larger dual.As an application, the authors characterize the largest subspace of H 1 , denoted by H b 1 with b ∈ BMO, such that the commutators [T, b] with classical sublinear operators T are bounded from H b 1 to L 1 . This endpoint boundedness of commutators allow the authors to give more applications. On the one hand, in the martingale setting, the authors obtain the endpoint estimates of commutators for both martingale transforms and martingale fractional integrals. On the other hand, in harmonic analysis, the authors establish the endpoint estimates of commutators both for the dyadic Hilbert transform beyond doubling measures and for the maximal operator of Cesàro means of Walsh-Fourier series.

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“…By proving that the dual space of H log (R n ) is the generalized BMO space BMO log (R n ) introduced by Nakai and Yabuta [50], which also characterizes the set of multipliers of BMO (R n ) (see also the recent survey [48] on this subject of Nakai), Bonami et al in [8] deduced that H log (R n ) in (1.3) is sharp in some sense. Moreover, Bonami et al in [7] proved that every atom of H log (R n ) can be written as a finite linear combination of product distributions in the space H 1 (R n ) × BMO (R n ) and, in dimension one, Bonami and Ky in [9] proved that H log (R) is indeed the smallest space satisfying (1.3). Recently, in [4,11], a bilinear decomposition theorem for multiplications of elements in H p (R n ) and its dual space C α (R n ) was established when p ∈ (0, 1) and α := 1/p − 1, and the sharpness of this bilinear decomposition was also obtained therein.…”

Section: Introductionmentioning

confidence: 99%

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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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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Atomic decomposition and weak factorization in generalized Hardy spaces of closed forms

Cited by 10 publications

References 19 publications

Real-variable characterizations of Orlicz-slice Hardy spaces

Real-variable characterizations of Orlicz-slice Hardy spaces

Products and Commutators of Martingales in $H_1$ and ${\rm BMO}$

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

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