Korean J. Math.  Vol 27, No 1 (2019)  pp.221-267
DOI: https://doi.org/10.11568/kjm.2019.27.1.221

The harmonic analysis associated to the Heckman-Opdam's theory and its application to a root system of type $BC_d$

Khalifa Trimeche

Abstract


In the five first sections of this paper we define and study the hypergeometric transmutation operators $V^W_k$ and ${}^tV^W_k$ called also the trigonometric Dunkl intertwining operator and its dual corresponding to the Heckman-Opdam's theory on $\mathbb{R}^d$. By using these operators we define the hypergeometric translation operator $\mathcal{T}^W_x, x \in \mathbb{R}^d$, and its dual ${}^t\mathcal{T}^W_x, x \in \mathbb{R}^d$, we express them in terms of the hypergeometric Fourier  transform $\mathcal{H}^W$, we give their properties and we deduce simple proofs of the Plancherel formula and the Plancherel theorem for the transform $\mathcal{H}^W$. We study also the hypergeometric convolution product on $W$-invariant $L^p_{\mathcal{A}_k}$-spaces, and we obtain some interesting results. In the sixth section we consider a some root system of type $BC_d$ (see [17]) of whom the corresponding hypergeometric translation operator is a positive integral operator. By using this positivity we improve the results of the previous sections and we prove others more general results.

Keywords


Cherednik's operators on $\mathbb{R}^d$ ; Heckman-Opdam's theory on $\mathbb{R}^d$; hypergeometric transmutation operators ; Hypergeometric translation operator ; Hypergeometric convolution product ; Heckman-Opdam's hypergeometric function ; Hypergeometri

Subject classification

33E30, 51F15, 33C67, 43A32, 43A62.

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