Un analogue d'un théorème de Hardy pour la transformation de Dunkl

Gallardo, Léonard; Trimèche, Khalifa

Comptes Rendus. Mathématique, Tome 334 (2002) no. 10, pp. 849-854.

On donne dans cette Note une généralisation d'un théorème de Hardy pour la transformation de Dunkl ${ℱ}_D$ sur ${\mathrm{R}}^d$. Plus précisément, pour toutes les valeurs de a>0, b>0 et p,q∈[1,+∞], on détermine les fonctions mesurables f telles que ${\mathrm{e}}^{{\mathrm{a}\left|\right|\mathrm{x}\left|\right|}^2}\mathrm{f}\in {\mathrm{L}}_k^p\left({\mathrm{R}}^d\right)$ et ${\mathrm{e}}^{{\mathrm{b}\left|\right|\mathrm{y}\left|\right|}^2}{ℱ}_D\left(\mathrm{f}\right)\in {\mathrm{L}}_k^q\left({\mathrm{R}}^d\right)$, où les ${\mathrm{L}}_k^p\left({\mathrm{R}}^d\right)$ sont les espaces L^p associés à la transformation de Dunkl.

In this Note we give a generalization of Hardy's theorem for the Dunkl transform ${ℱ}_D$ on ${\mathrm{R}}^d$. More precisely, for all a>0, b>0 and p,q∈[1,+∞], we determine the measurable functions f such that ${\mathrm{e}}^{{\mathrm{a}\left|\right|\mathrm{x}\left|\right|}^2}\mathrm{f}\in {\mathrm{L}}_k^p\left({\mathrm{R}}^d\right)$ and ${\mathrm{e}}^{{\mathrm{b}\left|\right|\mathrm{y}\left|\right|}^2}{ℱ}_D\left(\mathrm{f}\right)\in {\mathrm{L}}_k^q\left({\mathrm{R}}^d\right)$, where ${\mathrm{L}}_k^p\left({\mathrm{R}}^d\right)$ are the L^p spaces associated with the Dunkl transform.

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