Léonard Gallardo

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 .

A chaotic representation property of the multidimensional Dunkl processes

Dunkl processes are martingales as well as cadlag homogeneous Markov processes taking values in R d and they are naturally associated with a root system. In this paper we study the jumps of these processes, we describe precisely their martingale decompositions into continuous and purely discontinuous parts and we obtain a Wiener chaos decomposition of the corresponding L 2 spaces of these processes in terms of adequate mixed multiple stochastic integrals.

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.

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.

A multidimensional central limit theorem for random walks on hypergroups

Random walks on a product hypergroup of classical one-dimensional hypergroups are d -dimensional Markov chains which have a conditional increment with an asymptotically constant covariance matrix. We use this property and a central limit theorem for martingale differences, to show that such random walks suitably normalized converge to a standard Gaussian law N (0, o ) in R d with a covariance matrix o which is explicitly determined.

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.

Vitesse de fuite et comportement asymptotique du mouvement brownien sur les groupes de Lie nilpotents

SummaryLet β(t) be a brownian motion on a nilpotent simply connected Lie group

Riesz potentials of Radon measures associated to reflection groups

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