Journées à Orléans
Après-midi harmoniqueD. Békollé, B. Amri et S. Ben Said (org. J-P. Anker et A. Bonami)
Wednesday 16 December 2009 14:00 - Orléans - Salle de Séminaire
Résumé :
14h-14h45 : David Békollé (Université de Ngaoundéré, Cameroun) Titre : Toeplitz et au-delà : Opérateurs continus et compacts dans les espaces de Bergman L^p_a, 1\leq p <\infty. Résumé : This talk is a report on a recent work of Dieudonné Agbor (Ph. D student) and Edgar Tchoundja (postdoc). It concerns the problem of the determination of L^1 symbols whose associated (Bergman-)Toeplitz operators are bounded (resp. compact) on Bergman spaces L^p_a, 1\le p< \infty in the unit disc of the complex plane. For p>1, we shall give a new presentation and improvements of earlier results of N. Zorboska, J. Miao and D. Zheng. The presented results were also generalized to the unit ball of C^n. For p=1 the results are new and related to the reproducing kernel thesis and to earlier results of T. Yu, Z. Wu, R. Zhao and N. Zorboska. Part of the presented study can be done for general linear operators on Bergman spaces. 14h45-15h30 : Béchir AMRI (IPEIT, Tunis) Titre : L^p-boundedness of Riesz transform for the Dunkl transform Résumé : Nous résolvons une question posée par S. Thangavelyu and Y. Xu dans leur article "Riesz transforms and Riesz potentials for the Dunkl transform", J. Comp. Appl. Math. 199 (2007), 181-195. 15h30-16h : pause café 16h-16h45 : Salem BEN SAID (Université Henri Poincaré - Nancy 1) Titre : Interpolation entre deux différentes représentations minimales de deux groupes différents Résumé : The classical Fourier transform is one of the most basic objects in analysis; it may be understood as belonging to a one-parameter group of unitary operators on L^2(R^N), and this group may even be extended holomorphically to a semigroup (the Hermite semigroup) I(z) generated by the self-adjoint operator ∆−||x||^2. This is a holomorphic semigroup of bounded operators depending on a complex variable z in the complex right half-plane, viz. I(z+w)=I(z)I(w). The structure of this semigroup and its properties may be appreciated without any reference to representation theory, whereas the link itself is rich as was revealed beautifully by R. Howe in connection with the Schrödinger model of the Weil representation. The aim of this talk is to consider the Dunkl Laplacian ∆_k and to construct a deformation of the classical situation, namely, a generalization F_{k,a} of the Fourier transform, and the holomorphic semigroup I_{k,a}(z) with infinitesimal generator ||x||^{2−a}∆_k−||x||^a, acting on a concrete Hilbert space deforming L^2(R^N). We analyze these operators F_{k,a} and I_{k,a}(z) in the context of integral operators as well as representation theory. Particular attention will be given to the cases a=1 and a=2. This is joint work with T. Kobayashi and B. Ørsted.
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