Riesz transforms for Dunkl transform

Amri, Bechir; Sifi, Mohamed

doi:10.5802/ambp.312

Riesz transforms for Dunkl transform
[Transformées de Riesz associés à la transformée de Dunkl]

Amri, Bechir ¹ ; Sifi, Mohamed ²

¹ Department of Mathematics University of Tunis Preparatory Institute of Engineer Studies of Tunis 1089 Montfleury, Tunis, Tunisia
² Department of Mathematics University of Tunis El Manar Faculty of Sciences of Tunis 2092 Tunis El Manar, Tunis, Tunisia

Annales mathématiques Blaise Pascal, Tome 19 (2012) no. 1, pp. 247-262.

Résumé
Abstract

Dans cet article, nous étudions la bornitude des transformées de Riesz associées à la transformée de Dunkl sur les espaces $L^{p}$ , $1 < p < \infty$ .

In this paper we obtain the $L^{p}$ -boundedness of Riesz transforms for the Dunkl transform for all $1 < p < \infty$ .

DOI : 10.5802/ambp.312

Classification : 17B22, 32A55, 43A32, 42A45
Mots-clés : Dunkl transforms, Riesz Transforms, Singular integrals

Affiliations des auteurs :

Amri, Bechir ¹ ; Sifi, Mohamed ²

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Amri, Bechir; Sifi, Mohamed. Riesz transforms for Dunkl transform. Annales mathématiques Blaise Pascal, Tome 19 (2012) no. 1, pp. 247-262. doi : 10.5802/ambp.312. http://www.numdam.org/articles/10.5802/ambp.312/

Bibliographie
Cité par

[1] Amri, B.; Gasmi, A.; Sifi, M. Linear and bilinear multiplier operators for the Dunkl transform, Mediterranean Journal of Mathematics, Volume 7 (2010), pp. 503-521 | DOI | MR | Zbl

[2] Dai, F.; Wang, H. A transference theorem for the Dunkl transform and its applications, Journal of Functional Analysis, Volume 258 (2010), pp. 4052-4074 | DOI | MR | Zbl

[3] Dunkl, C. F. Differential–Difference operators associated to reflection groups, Trans. Amer. Math., Volume 311 (1989), pp. 167-183 | DOI | MR | Zbl

[4] Hassani, S.; Mustapha, S.; Sifi, M. Riesz potentials and fractional maximal function for the Dunkl transform, J. Lie Theory, Volume 19 (2009, no. 4), pp. 725-734 | MR | Zbl

[5] de Jeu, M.F.E. The Dunkl transform, Invent. Math., Volume 113 (1993), pp. 147-162 | DOI | EuDML | MR | Zbl

[6] Rosler, M. Dunkl operators: theory and applications, in Orthogonal polynomials and special functions (Leuven, 2002), $ℝ^{N}$ , Lect. Notes Math., Volume 1817 (2003), pp. 93-135 | MR | Zbl

[7] Rosler, M. A positive radial product formula for the Dunkl kernel, Trans. Amer. Math. Soc., Volume 355 (2003), pp. 2413-2438 | DOI | MR | Zbl

[8] Stein, E. M. Harmonic Analysis: Reals-Variable Methods, Orthogonality and Oscillatory Integrals, PrincetonS, New Jersey, 1993 | MR | Zbl

[9] Thangavelu, S.; Xu, Y. Convolution operator and maximal function for Dunkl transform, J. Anal. Math., Volume 97 (2005), pp. 25-55 | DOI | MR | Zbl

[10] Thangavelu, S.; Xu, Y. Riesz transforms and Riesz potentials for the Dunkl transform, J. Comp. and Appl. Math., Volume 199 (2007), pp. 181-195 | DOI | MR

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