Khalifa Trimèche

Positivity of the Jacobi–Cherednik intertwining operator and its dual

Abstract This paper is devoted to the study of the differential-difference Jacobi–Cherednik operator defined for f ∈ C 1 (𝕣) by T (k,k′) f(x) = f′(x) + (k coth(x) + k′ tanh(x))(f(x) – f(–x)) – (k + k′) f(–x), where k > 0 and k′ ≥ 0 are two parameters, and to the positivity of the operator which intertwines T (k,k′) and the derivative operator .

An L P version of Hardy's theorem for the Dunkl transform

In this paper, we give a generalization of Hardys theorems for the Dunkl transform ℱ D on ℝ d . More precisely for all a > 0, b > 0 and p, q ∈ [1, + ∞], we determine the measurable functions f on ℝ d such that where are the Lebesgue spaces associated with the Dunkl transform.

Harmonic analysis associated with the Cherednik operators and the Heckman–Opdam theory

Abstract By using the trigonometric Dunkl intertwining operator and its dual introduced by the author in [Trimèche, Adv. Pure Appl. Math.], we define and study the hypergeometric translation operators associated with the Cherednik operators Tj , j = 1, 2, . . . , d. Next with the help of these translation operators, we define and study the hypergeometric convolution product of functions and distributions associated with the operators Tj , j = 1, 2, . . . , d.

The trigonometric Dunkl intertwining operator and its dual associated with the Cherednik operators and the Heckman–Opdam theory

Abstract In this paper we prove that there exists a unique topological isomorphism Vk from (the space of C∞ -functions on ) onto itself which intertwines the Cherednik operators Tj, j = 1, 2, . . . , d, and the partial derivatives , j = 1, 2, . . . , d, called the trigonometric Dunkl intertwining operator (this name has been proposed by G. J. Heckman). To define and study the operator Vk we have introduced first the trigonometric Dunkl dual intertwining operator t Vk . The operators Vk and t Vk are the analogue in the Dunkl theory of the Dunkl intertwining operator and its dual (see [Dunkl, Can. J. Math. 43: 1213–1227, 1991, Trimèche, Integrals Transforms Special Funct. 12: 349–374, 2001]).

Un analogue d'un théorème de Hardy pour la transformation de Dunkl

In this Note we give a generalization of Hardys theorem for the Dunkl transform FD on Rd. More precisely, for all a>0, b>0 and p,q∈[1,+∞], we determine the measurable functions f such that ea||x||2f∈Lkp(Rd) and eb||y||2FD(f)∈Lkq(Rd), where Lkp(Rd) are the Lp spaces associated with the Dunkl transform. To cite this article: L. Gallardo, K. Trimeche, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 849–854.

Positivity of the transmutation operators associatedwith a Cherednik type operator on the real line

Abstract. In this paper we prove the positivity of the transmutation operators associated with a Cherednik type operator on the real line, and we deduce the positivity of the transmutation operators related to the Jacobi–Cherednik operator on ℝ and the Cherednik operator in the one-dimensional case.

L'équation de Poisson et les noyaux de Green associés à un opérateur différentiel singulier sur R+

We consider a class ℳ of singular differential operators on the half line and ⋆ the convolution on ℝ+. associated with L e ℳ. If μ(≠ ɛ0) is a probability measure on ℝ+, we study the asymptotic behaviour of the solution of both Poisson equations Lu = −ƒ and (μ. − ɛ0) ⋆ u = −ƒ where ƒ e Ck(ℝ+) is given. The results follow from a more general study on the precise asymptotic behaviour of the Green kernel of the convolution semigroups associated with L.

The positivity of the hypergeometric translation operators associated to the Cherednik operators and the Heckman--Opdam theory on ℝ d

Abstract We consider the hypergeometric translation operators associated to the Cherednik operators and the Heckman–Opdam theory on ℝ d

The positivity of the transmutation operators associated with theCherednik operatorsattached to the root system of type A2

{\mathbb{R}^{d}}

Absolute continuity of the representing measures of the Dunkl intertwining operator and of its dual and applications

introduced by the author in [6]. Under some conditions on the root system and the multiplicity function, we prove in this paper that these operators are positivity preserving and allow positive integral representations. In particular, we deduce for the Opdam–Cherednik kernel and the Heckman–Opdam hypergeometric function the following main results: (i) Their product formulas are positive integral transforms. (ii) We obtain for them best estimates.