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Theses

Analyse harmonique associée à des systèmes de racines et aux opérateurs de Dunkl rationnels

Abstract : In this thesis, we focus on harmonic analysis and special functions associated with rational Dunkl operators which are deformations by reflections of directional derivatives. They provide an essential tool to extend, in the setting of root systems and associated reflection groups, Fourier analysis on Euclidean spaces and analysis on Riemannian symmetric spaces of Euclidean type. After a detailed survey on Dunkl theory, we study the maximal operator defined in this setting. We begin with some improvements on the behavior of the constants of the maximal theorem proved by Thangavelu and Xu for a general reflection group. We then extend their theorem to the vector-valued case by establishing Fefferman-Stein inequalities in the case Z_2^d. In order to prove this theorem, we construct an operator of Hardy-Littlewood type on which, contrary to the Dunkl maximal operator, we can apply the methods of real analysis. In particular, we give a sharp estimate of the generalised translation of the characteristic function of a ball. Afterwards, our study is devoted to some results on exponential integrability which completes the Fefferman-Stein inequalities, and to a vector-valued maximal theorem for Bessel-Kingman hypergroups. Finally, we develop the Dunkl analysis in the case of a positive subsystem of orthogonal roots. We give an explicit formula of the Dunkl kernel and a product formula which implies that the Dunkl translation is a bounded operator. We also focus on the particular case of a root system of A_1 type in odrer to give an equality which links normalized Bessel functions and Gegenbauer polynomials.
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Submitted on : Monday, January 24, 2011 - 10:04:43 AM
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Luc Deleaval. Analyse harmonique associée à des systèmes de racines et aux opérateurs de Dunkl rationnels. Mathématiques [math]. Université Pierre et Marie Curie - Paris VI, 2010. Français. ⟨tel-00558751⟩

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