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Pré-Publication, Document De Travail Année : 2016

$L^p$-Fourier analysis associated to a family of differential-reflection operators

Salem Ben Said
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Mohamed Sifi
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  • PersonId : 859766

Résumé

In a previous paper we introduced a family of differential-reflection operators $\Lambda_{A, \varepsilon}$ acting on smooth functions defined on $\mathbb R,$ where the spectral problem for the operators $\Lambda_{A, \varepsilon}$ has been discussed. Here $A$ is a Sturm-Liouville function with additional hypotheses and $\varepsilon\in \mathbb R.$ Via the eigenfunctions of $\Lambda_{A,\varepsilon},$ we introduce in this paper a generalized Fourier transform $\mathcal F_{A,\varepsilon}.$ An $L^p$-harmonic analysis for $\mathcal F_{A,\varepsilon}$ is developed when $0< p \leq \frac{2}{1+\sqrt{1-\varepsilon^2}}$ and $-1\leq \varepsilon\leq 1.$ In particular, an $L^p$-Schwartz space isomorphism theorem for $\mathcal F_{A,\varepsilon}$ is proved.
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Dates et versions

hal-01282503 , version 1 (03-03-2016)

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  • HAL Id : hal-01282503 , version 1

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Salem Ben Said, Asma Boussen, Mohamed Sifi. $L^p$-Fourier analysis associated to a family of differential-reflection operators. 2016. ⟨hal-01282503⟩
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