Multi-parameter Triebel-Lizorkin and Besov spaces associated with flag singular integrals

Ding, Yong; Lu, Guo Zhen; Lin, Benyao

doi:10.1007/s10114-010-8352-8

Cited by 28 publications

(7 citation statements)

References 18 publications

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“…For more results related to multi-parameter theory, see [1,2,3,4,10,11,12,13,15,16,17,18,20,22,23,24,27,28,29,31,33,34,35,36]. Recently, the authors of [9,25] established the boundedness of singular integral operators on multi-parameter Triebel-Lizorkin and Besov-Lipschitz spaces. More recently, the atomic decomposition and dual spaces for multi-parameter Triebel-Lizorkin spaces associated with the composition of two singular operators studied by Phong and Stein [32] were given in [7,6].…”

Section: Introduction and Statement Of Main Resultsmentioning

confidence: 99%

Boundedness of multi-parameter Fourier multiplier operators on Triebel–Lizorkin and Besov–Lipschitz spaces

Chen

Liu

2016

Nonlinear Analysis

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The main purpose of this paper is threefold. First, we prove that under the limited smoothness conditions, multi-parameter Fourier multiplier operators are bounded on multi-parameter Triebel-Lizorkin and Besov-Lipschitz spaces by the Littlewood-Paley decomposition and the strong maximal operator. Second, we offer a different and more direct method to deal with the boundedness instead of transforming Fourier multiplier operators into multiparameter Calderón-Zygmund operators. Third, we also prove the boundedness of multi-parameter Fourier multiplier operators on weighted multiparameter Triebel-Lizorkin and Besov-Lipschitz spaces when the Fourier multiplier is only assumed with limited smoothness.

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Section: Introduction and Statement Of Main Resultsmentioning

confidence: 99%

Boundedness of multi-parameter Fourier multiplier operators on Triebel–Lizorkin and Besov–Lipschitz spaces

Chen

Liu

2016

Nonlinear Analysis

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“…These tools and geometry do not apply in the flag setting on H ν . Further progress on related problems in the flag setting includes (but is not limited to) [13,27,30,31,59,64,65].…”

Section: Background and Questionsmentioning

confidence: 99%

Flag Hardy space theory on Heisenberg groups and applications

Chen¹,

Cowling²,

Lee³

et al. 2021

Preprint

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We establish a complete theory of the flag Hardy space H 1 F (H ν ) on the Heisenberg group H ν with characterisations via atomic decompositions, area functions, square functions, maximal functions and singular integrals. We introduce several new techniques to overcome the difficulties caused by the noncommutative Heisenberg group multiplication, and the lack of a suitable Fourier transformation and Cauchy-Riemann type equations. Applications include the boundedness fromvarious singular integral operators that arise in complex analysis, a sharp boundedness result on H 1 F (H ν ) of the Marcinkiewicz-type multipliers introduced by Müller, Ricci and Stein, and the decomposition of flag BMO space via singular integrals. Contents 3.7 The inclusion H 1 F,atom (H ν ) ⊆ H 1 F,gmax (H ν

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“…Inspired by those works, and recent study of multi-parameter Triebel-Lizorkin spaces in the pure product setting and as well as in the setting associated with the ag singular integrals [9,29], we consider multi-parameter Triebel-Lizorkin spaces associated with the composition of two singular integral operators of di erent homogeneities. In particular, we are interested in the atomic decomposition of these spaces.…”

Section: Introductionmentioning

confidence: 97%

Multi-parameter Triebel–Lizorkin spaces associated with the composition of two singular integrals and their atomic decomposition

Ding

Zhu

2014

Forum Mathematicum

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Atomic decomposition plays an important role in establishing the boundedness of operators on function spaces. Let 0 < , < ∞ and = ( 1 , 2 ) ∈ ℝ 2 . In this paper, we introduce multi-parameter Triebel-Lizorkin spaceṡ , (ℝ ) associated with di erent homogeneities arising from the composition of two singular integral operators whose weak (1, 1) boundedness was rst studied by Phong and Stein [32]. We then establish its atomic decomposition which is substantially di erent from that for the classical one-parameter Triebel-Lizorkin spaces. As an application of our atomic decomposition, we obtain the necessary and su cient conditions for the boundedness of an operator on the multi-parameter Triebel-Lizorkin type spaces. In the special case of 1 = 2 = 0, = 2 and 0 < ≤ 1, our spaceṡ , (ℝ ) coincide with the Hardy spaces com associated with the composition of two di erent singular integrals (see [19]). Therefore, our results also give an atomic decomposition of com . Our work appears to be the rst result of atomic decomposition in the Triebel-Lizorkin spaces in the multi-parameter setting.

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Multi-parameter Triebel-Lizorkin and Besov spaces associated with flag singular integrals

Cited by 28 publications

References 18 publications

Boundedness of multi-parameter Fourier multiplier operators on Triebel–Lizorkin and Besov–Lipschitz spaces

Boundedness of multi-parameter Fourier multiplier operators on Triebel–Lizorkin and Besov–Lipschitz spaces

Flag Hardy space theory on Heisenberg groups and applications

Multi-parameter Triebel–Lizorkin spaces associated with the composition of two singular integrals and their atomic decomposition

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