Uncertainty principles and characterization of the heat kernel for certain differential-reflection operators

Saïd, Salem Ben; Boussen, Asma; Sifi, Mohamed

doi:10.1515/apam-2015-5010

Cited by 3 publications

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References 24 publications

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“…For ε = 0 and 1, this result can be found in [37] and [38], while for A(x) = A α (x) = |x| 2α+1 and ε arbitrary, the positivity of V A,ε goes back to [28]. We pin down that the positivity of V A,ε played a fundamental role in [3] in establishing an analog of Beurling's theorem and its relatives, such as theorems of the type Gelfand-Shilov, Morgan's, Hardy's, and Cowling-Price in the setting of this paper.…”

Section: Introductionmentioning

confidence: 67%

Intertwining Operators Associated to a Family of Differential-Reflection Operators

2016

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We introduce a family of differential-reflection operators ΛA,ε acting on smooth functions defined on R. Here, A is a Sturm-Liouville function with additional hypotheses and −1 ≤ ε ≤ 1. For special pairs (A, ε), we recover Dunkl's, Heckman's and Cherednik's operators (in one dimension). The spectral problem for the operators ΛA,ε is studied. In particular, we obtain suitable growth estimates for the eigenfunctions of ΛA,ε. As the operators ΛA,ε are a mixture of d/dx and reflection operators, we prove the existence of an intertwining operator VA,ε between ΛA,ε and the usual derivative. The positivity of VA,ε is also established.Mathematics Subject Classification. 34K99, 34B25, 33E30.

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Section: Introductionmentioning

confidence: 67%

Intertwining Operators Associated to a Family of Differential-Reflection Operators

2016

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“…This convolution is commutative, associative and its satisfies the following properties, for the proofs we refer the reader to [2,24,25]. Proposition 2.5.…”

Section: The Translation Operator Associated With the Jacobi-dunkl Tr...mentioning

confidence: 99%

“…The following proposition summarizes some properties of the Jacobi-Dunkl translation operator, for the proofs we refer the reader to [1,2,24,25] . Proposition 2.4.…”

Section: The Translation Operator Associated With the Jacobi-dunkl Tr...mentioning

confidence: 99%

Spectral theorems for Wigner transform associated with the Jacobi-Dunkl localization operators

Chana

Akhlidj

2023

Preprint

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The main purpose of this paper is to define the Wigner transform associated with the Jacobi-Dunkl operator and to give some results related to this transform as inversion formula, next motivated by Wong’s point of view we define a class of pseudo-differential operator Lu,v(σ) called localization operator wich depend on a symbol σ and two functions u and v, we give a criteria in terms of the symbol σ for its boundedness and compactness,we also show that these operators belongs to the Schatten-Von Neumann class Sp for all p ∈ [1; +∞] and we give a trace formula. MSC2020-Mathematics Subject Classification: 42B10, 47G30, 47B10.

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Lp harmonic analysis for differential-reflection operators

Saïd

Boussen

Sifi

2016

Comptes Rendus. Mathématique

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Uncertainty principles and characterization of the heat kernel for certain differential-reflection operators

Abstract: AbstractWe prove various versions of uncertainty principles for a certain Fourier transform ℱ

Cited by 3 publications

References 24 publications

Intertwining Operators Associated to a Family of Differential-Reflection Operators

Intertwining Operators Associated to a Family of Differential-Reflection Operators

Spectral theorems for Wigner transform associated with the Jacobi-Dunkl localization operators

Lp harmonic analysis for differential-reflection operators

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