Endpoint for the div-curl lemma in Hardy spaces

Bonami, Aline; Feuto, Justin; Grellier, Sandrine

doi:10.5565/publmat_54210_03

Cited by 61 publications

(36 citation statements)

References 17 publications

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“…We should also point out that the space H log (R n ) arises naturally in the study of pointwise product of functions in H 1 (R n ) with functions in BMO(R n ), and in the endpoint estimates for the div-curl lemma (see for example [3,4,23]). …”

Section: Some Preliminaries and Notationsmentioning

confidence: 99%

Bilinear decompositions and commutators of singular integral operators

2012

Trans. Amer. Math. Soc.

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Abstract. Let b be a BM O-function. It is well-known that the linear commutator [b, T ] of a Calderón-Zygmund operator T does not, in general, map continuouslyIn this paper, we find the largest subspace H 1 b (R n ) such that all commutators of Calderón-Zygmund operators are continuous fromWe also study the commutators [b, T ] for T in a class K of sublinear operators containing almost all important operators in harmonic analysis. When T is linear, we prove that there exists a bilinear operators R = R T mapping continuouslywhere S is a bounded bilinear operator fromIn the particular case of T a Calderón-Zygmund operator satisfying T 1 = T * 1 = 0 and b in BM O log (R n )-the generalized BMO type space that has been introduced by Nakai and Yabuta to characterize multipliers of BMO(R n ) -we prove that the commutatorWhen T is sublinear, we prove that there exists a bounded subbilinear operator R = R T :The bilinear decomposition (1) and the subbilinear decomposition (2) allow us to give a general overview of all known weak and strong L 1 -estimates.

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Section: Some Preliminaries and Notationsmentioning

confidence: 99%

Bilinear decompositions and commutators of singular integral operators

2012

Trans. Amer. Math. Soc.

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“…On a donné dans la section 4.2.4 une version limite d'un lemme div-curl, dans laquelle on suppose le champ eà divergence nulle et L ∞ et le champ Fà rotationnel nul dans un espace de Hardy sur R n ou sur un domaine, et la conclusion est que e · F appartientà un espace de Hardy. Si on suppose maintenant le champ e dans bmo(R n , R n ) (version locale de BM O, voir la section 2.3.6), et toujours F dans H 1 (R n , R n )à rotationnel nul, on peut encore donner un sens au produit e · F , et ce produit appartient alorsà un espace de Hardy-Orlicz sur R n ( [59]). Des versions plus générales de ce résultat pour des formes différentielles sontégalement données dans [59].…”

Section: Compléments Et Perspectives Sur Les Lemmes Div-curlunclassified

“…Si on suppose maintenant le champ e dans bmo(R n , R n ) (version locale de BM O, voir la section 2.3.6), et toujours F dans H 1 (R n , R n )à rotationnel nul, on peut encore donner un sens au produit e · F , et ce produit appartient alorsà un espace de Hardy-Orlicz sur R n ( [59]). Des versions plus générales de ce résultat pour des formes différentielles sontégalement données dans [59]. Donner desénoncés analogues sur des domaines de R n est une question ouverte.…”

Section: Compléments Et Perspectives Sur Les Lemmes Div-curlunclassified

Racines Carrées d'OPÉRATEURS Elliptiques Et Espaces De Hardy

Russ

2011

Confluentes Math.

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“…Here, A q (R n ) with q ∈ [1, ∞] denotes the class of Muckenhoupt weights (see, for example, [11,12,13] for their definitions and properties). Moreover, the space H ϕ (R n ) also has already found many applications in analysis (see, for example, [3,4,17,22,23] and their references).…”

Section: Theorem 11 ([9]mentioning

confidence: 99%

“…Also, the Musielak-Orlicz-Hardy space H ϕ (R n ) has proved useful in the study of other analysis problems when we take various different Musielak-Orlicz functions ϕ (see, for example, [3,22,23]). …”

Section: Definition 12 ([23]mentioning

confidence: 99%

Riesz transform characterizations of Musielak-Orlicz-Hardy spaces

Cao

Chang

Yang

et al. 2016

Trans. Amer. Math. Soc.

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Abstract. Let ϕ be a Musielak-Orlicz function satisfying that, for any (x, t) ∈ R n × (0, ∞), ϕ(·, t) belongs to the Muckenhoupt weight class A ∞ (R n ) with the critical weight exponent q(ϕ) ∈ [1, ∞) and ϕ(x, ·) is an Orlicz function with 0 < i(ϕ) ≤ I(ϕ) ≤ 1 which are, respectively, its critical lower type and upper type. In this article, the authors establish the Riesz transform characterizations of the Musielak-Orlicz-Hardy spaces H ϕ (R n ) which are generalizations of weighted Hardy spaces and Orlicz-Hardy spaces. Precisely, the authors characterize H ϕ (R n ) via all the first order Riesz transforms whenn , and via all the Riesz transforms with the order not more than m ∈ N when i(ϕ) q(ϕ) > n−1 n+m−1 . Moreover, the authors also establish the Riesz transform characterizations of H ϕ (R n ), respectively, by means of the higher order Riesz transforms defined via the homogenous harmonic polynomials or the odd order Riesz transforms. Even if when ϕ(x, t) := tw(x) for all x ∈ R n and t ∈ [0, ∞), these results also widen the range of weights in the known Riesz characterization of the classical weighted Hardy space H 1 w (R n ) obtained by R. L. Wheeden from w ∈ A 1 (R n ) into w ∈ A ∞ (R n ) with the sharp range q(w) ∈ [1, n n−1 ), where q(w) denotes the critical index of the weight w.

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Endpoint for the div-curl lemma in Hardy spaces

Cited by 61 publications

References 17 publications

Bilinear decompositions and commutators of singular integral operators

Bilinear decompositions and commutators of singular integral operators

Racines Carrées d'OPÉRATEURS Elliptiques Et Espaces De Hardy

Riesz transform characterizations of Musielak-Orlicz-Hardy spaces

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