Pseudo-multipliers and Smooth Molecules on Hermite Besov and Hermite Triebel–Lizorkin Spaces

Ly, Fu Ken; Naibo, Virginia

doi:10.1007/s00041-021-09856-9

Cited by 6 publications

(15 citation statements)

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“…It turns out that these spaces are independent of the choice of ϕ; they are also in general different from the classical Triebel-Lizorkin and Besov spaces associated to the Laplacian operator in R n . For the details (as well as other related facts) see [8,14,30,37] and earlier works cited there. Throughout the rest of this article we will adopt the following notational conventions.…”

Section: Hermite Function Spaces and Their Decompositionsmentioning

confidence: 99%

“…In this section, we define Hermite Besov and Hermite Triebel-Lizorkin spaces and describe two decompositions, one through frames and one through smooth molecules. The full details and background can be found in [14,30,37] (see also [9,12,13]).…”

Section: Hermite Function Spaces and Their Decompositionsmentioning

confidence: 99%

“…They are constructed from the zeros of Hermite polynomials and we provide a brief description of their geometry here. Further details can be found in [14,30,37].…”

Section: Hermite Function Spaces and Their Decompositionsmentioning

confidence: 99%

“…where e N (x) := e −c|x| 2 for some c > 0 if |x| 2 ≤ N , and e N (x) := 1 otherwise. For further details and related estimates see [30,Lemma A.1] and [14, Propostion 2.2].…”

Section: Hermite Function Spaces and Their Decompositionsmentioning

confidence: 99%

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Hermite Besov and Triebel–Lizorkin spaces and applications

Ly,

Naibo

2023

Revista De La Unión Matemática Argentina

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Section: Hermite Function Spaces and Their Decompositionsmentioning

confidence: 99%

Section: Hermite Function Spaces and Their Decompositionsmentioning

confidence: 99%

“…They are constructed from the zeros of Hermite polynomials and we provide a brief description of their geometry here. Further details can be found in [14,30,37].…”

Section: Hermite Function Spaces and Their Decompositionsmentioning

confidence: 99%

“…where e N (x) := e −c|x| 2 for some c > 0 if |x| 2 ≤ N , and e N (x) := 1 otherwise. For further details and related estimates see [30,Lemma A.1] and [14, Propostion 2.2].…”

Section: Hermite Function Spaces and Their Decompositionsmentioning

confidence: 99%

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Hermite Besov and Triebel–Lizorkin spaces and applications

Ly,

Naibo

2023

Revista De La Unión Matemática Argentina

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“…This type of cancellation condition was further used by the first author and Liflyand [8] to give a molecular decomposition and prove Goldberg's version of Hardy's inequality (see [13]) for h 1 (R). Approximate moments conditions have also been considered very recently by Ly and Naibo [19] in the Hermite setting.…”

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confidence: 99%

Inhomogeneous cancellation conditions and Calderón-Zygmund type operators on $h^p$

Dafni¹,

Lau²,

Picon³

et al. 2021

Preprint

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A. In this work we present a new approach to molecules on Goldberg's local Hardy spaces h p (R n ), 0 < p ≤ 1, assuming an appropriate cancellation condition. As applications, we prove a version of Hardy's inequality and improved continuity results for inhomogeneous Calderón-Zygmund operators on these spaces. IAtomic decomposition in the Hardy spaces H p (R n ), 0 < p ≤ 1, allows us to write any tempered distribution f ∈ H p (R n ) as an infinite linear combination j λ j a j of atoms a j , with f H p ≈ inf ( |λ j | p ) 1/p over all such decompositions. An atom is a function supported in a ball (or a cube) which satisfies a size condition relative to that ball, and vanishing moment conditions. Several properties and applications of H p (R n ) follow from this important tool; for instance, if T : S ′ (R n ) → S ′ (R n ) is a linear and continuous operator then its extension and continuity on H p (R n ) can be established by just verifying that T a j H p ≤ C uniformly (see [4,20] for other cases). In the particular case when T is a convolution-type singular integral operator, the uniform control on the atoms can be verified (see [12, Chapter III p. 325]) and in addition, M j := T a j are molecules. In contrast to atoms, molecules do not require compact support, but satisfy a special concentration of Lebesgue norm associated to some ball, as well as vanishing moment conditions. The molecular theory on Hardy spaces was first studied by Coifman [5] in order to characterize the Fourier transform of distributions on H p (R), and by Coifman, Taibleson and Weiss in the subsequent works [6,24]. This theory has been extensively explored in general settings, in particular to study the continuity of certain classes of non-convolution Calderón-Zygmund operators and their generalizations (see [2]) on Hardy spaces, as well as on more general function spaces such as Triebel-Lizorkin (see [25]).The spaces H p (R n ) for 0 < p ≤ 1 are not closed under multiplication by test functions, since this may destroy the global vanishing moment conditions. Consequently, we do not expect that linear operators such as non-convolution operators (for instance pseudodifferential operators) maps H p (R n ) to itself. For this reason, Goldberg [14] introduced a localizable or inhomogeneous version of Hardy spaces, which he called local Hardy spaces and denoted by h p (R n ), satisfying the continuous inclusions [14, Lemma 4]), a natural atomic decomposition for h p (R n ) arises which requires vanishing moment conditions only for atoms supported on balls B with radius r(B) ≤ 1; for atoms with r(B) > 1, no moment conditions are required. As an application, Goldberg shows that pseudodifferential operators in the class OpS 0 1,0 (R n ) are bounded on h p (R n ) (see [16] for general symbols in Hörmander classes).In contrast to H p (R n ), the molecular theory for h p (R n ), 0 < p ≤ 1, is still not completely well understood. The initial formulation by Komori [17], for n/(n + 1) < p < 1, was used to obtain boundedness of standard Calderón-Zygmund oper...

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Weighted estimates for Hermite pseudo-multipliers with rough symbols

2024

Journal of Approximation Theory

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Pseudo-multipliers and Smooth Molecules on Hermite Besov and Hermite Triebel–Lizorkin Spaces

Cited by 6 publications

References 33 publications

Hermite Besov and Triebel–Lizorkin spaces and applications

Hermite Besov and Triebel–Lizorkin spaces and applications

Inhomogeneous cancellation conditions and Calderón-Zygmund type operators on $h^p$

Weighted estimates for Hermite pseudo-multipliers with rough symbols

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