New Hardy Spaces of Musielak–Orlicz Type and Boundedness of Sublinear Operators

Ky, Luong Dang

doi:10.1007/s00020-013-2111-z

Cited by 153 publications

(365 citation statements)

References 43 publications

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“…Furthermore, it is also shown in [23] that the space BMO log (R n ) is the dual of the space H log (R n ).…”

Section: Some Preliminaries and Notationsmentioning

confidence: 98%

“…The space H log (R n ) (see [23,4]) denotes the space of all tempered distributions f such that Mf ∈ L log (R n ) equipped with the norm…”

Section: Some Preliminaries and Notationsmentioning

confidence: 99%

“…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%

“…Remark that the interest of dealing with finite atomic decompositions has been underlined recently, for instance in [34,23]. Now, we denote by H 1 fin (R n ) the vector space of all finite linear combinations of ψ-atoms, that is,…”

Section: Theorem 22 (Atomic Decomposition)mentioning

confidence: 99%

“…Let f ∈ H 1 b (R n ), we prove [b, T ](f ) ∈ h 1 (R n ) using the fact that BMO log (R n ) is the dual of H log (R n ) (see [23]). Indeed, by Theorem 2.2, there exists a decomposition f = ∞ j=1 I⊂R j σ∈E f, ψ σ I ψ σ I where I⊂R j σ∈E f, ψ σ I ψ σ I are multiples of ψ-atoms related to the dyadic cubes Lemma 7.3), and thus …”

Section: Proof Of Theorem 32mentioning

confidence: 99%

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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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“…Furthermore, it is also shown in [23] that the space BMO log (R n ) is the dual of the space H log (R n ).…”

Section: Some Preliminaries and Notationsmentioning

confidence: 98%

“…The space H log (R n ) (see [23,4]) denotes the space of all tempered distributions f such that Mf ∈ L log (R n ) equipped with the norm…”

Section: Some Preliminaries and Notationsmentioning

confidence: 99%

Section: Some Preliminaries and Notationsmentioning

confidence: 99%

Section: Theorem 22 (Atomic Decomposition)mentioning

confidence: 99%

Section: Proof Of Theorem 32mentioning

confidence: 99%

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Bilinear decompositions and commutators of singular integral operators

2012

Trans. Amer. Math. Soc.

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Bilinear decompositions for the product space

2014

Mathematische Nachrichten

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In this paper, we improve a recent result by Li and Peng on products of functions in HL1double-struckRd and italicBMOLdouble-struckRd, where L=−Δ+V is a Schrödinger operator with V satisfying an appropriate reverse Hölder inequality. More precisely, we prove that such products may be written as the sum of two continuous bilinear operators, one from HL1double-struckRd×italicBMOLdouble-struckRd into L1double-struckRd, the other one from HL1double-struckRd×italicBMOLdouble-struckRd into Hprefixlogdouble-struckRd, where the space Hprefixlogdouble-struckRd is the set of distributions f whose grand maximal function Mf satisfies ∫Rd|Mf(x)|log(e+|Mf(x)|)+log(e+|x|)dx<∞.

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Weak Musielak–Orlicz Hardy spaces and applications

Liang

Yang

Jiang

2015

Mathematische Nachrichten

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Let φ:Rn×[0,∞)→[0,∞) satisfy that φ(x,·), for any given x∈Rn, is an Orlicz function and φ(·,t) is a Muckenhoupt A∞(double-struckRn) weight uniformly in t∈(0,∞). In this article, the authors introduce the weak Musielak–Orlicz Hardy space WHφ(double-struckRn) via the grand maximal function and then obtain its vertical or its non–tangential maximal function characterizations. The authors also establish other real‐variable characterizations of WHφ(Rn), respectively, in terms of the atom, the molecule, the Lusin area function, the Littlewood–Paley g‐function or gλ*‐function. All these characterizations for weighted weak Hardy spaces WHwp(double-struckRn) (namely, φ(x,t):=w(x)tp for 0.33em all t∈[0,∞) and x∈Rn with p∈(0,1] and w∈A∞(Rn)) are new and part of these characterizations even for weak Hardy spaces WHp(double-struckRn) (namely, φ(x,t):=tp for 0.33em all t∈[0,∞) and x∈Rn with p∈(0,1]) are also new. As an application, the boundedness of Calderón–Zygmund operators from Hφ(Rn) to WHφ(double-struckRn) in the critical case is presented.

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New Hardy Spaces of Musielak–Orlicz Type and Boundedness of Sublinear Operators

Cited by 153 publications

References 43 publications

Bilinear decompositions and commutators of singular integral operators

Bilinear decompositions and commutators of singular integral operators

Bilinear decompositions for the product space

Weak Musielak–Orlicz Hardy spaces and applications

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