# $L^p$ harmonic analysis for differential-reflection operators

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1 Analyse IECL - Institut Élie Cartan de Lorraine 2 Analyse Mathématique et Applications

Résumé : We introduce and study differential-reflection operators$\Lambda {A, \varepsilon}$ acting on smooth functions defined on $\mathbb R.$ Here $A$ is a Sturm-Liouville function with additional hypotheses and $\varepsilon\in \mathbb R.$ For special pairs $A,\varepsilon,$ we recover Dunkl-s, Heckman-s and Cherednik-s operators in one dimension. As, by construction, the operators $\Lambda {A, \varepsilon}$ are mixture of ${ m d}-{ m d}x$and reflection operators, we prove the existence of an operator $V {A,\varepsilon}$ so that $\Lambda {A, \varepsilon}\circ V {A,\varepsilon}=V {A,\varepsilon}\circ { m d}-{ m d}x.$ The positivity of the intertwining operator $V {A,\varepsilon}$ is also established. Via the eigenfunctions of $\Lambda {A,\varepsilon},$ we introduce a generalized Fourier transform $\mathcal F {A,\varepsilon}.$ For $-1\leq \varepsilon\leq 1$ and $0 < p \leq \frac{2}{1+\sqrt{1-\varepsilon^2}},$ we develop an $L^p$-Fourier analysis for $\mathcal F {A,\varepsilon},$ and then we prove an $L^p$-Schwartz space isomorphism theorem for $\mathcal F {A,\varepsilon}$.Details of this paper will be given in another article.

Autor: Salem Ben Saïd - Asma Boussen - Mohamed Sifi -

Fuente: https://hal.archives-ouvertes.fr/

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