Korean J. Math. Vol. 24 No. 4 (2016) pp.693-722
DOI: https://doi.org/10.11568/kjm.2016.24.4.693

Spectral theorems associated to the Dunkl operators

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Hatem Mejjaoli


In this paper, we characterize the support for the Dunkl transform on the generalized Lebesgue spaces via the Dunkl resolvent function. The behavior of the sequence of $L^{p}_{k}-$norms of iterated Dunkl potentials is studied depending on the support of their Dunkl transform. We systematically develop real Paley-Wiener theory for the Dunkl transform on $\mathbb{R}^{d}$ for distributions, in an elementary treatment based on the inversion theorem. Next, we improve the Roe's theorem associated to the Dunkl operators.

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[1] N.B. Andersen and M.F.E. de Jeu, Real Paley-Wiener theorems and local spectral radius formulas, Trans. Amer. Math. Soc. 362 (2010), 3616–3640. Google Scholar

[2] N.B. Andersen, Roe’s theorem revisited, Integral Transf. and Special Functions V. 26 Issue 3, (2015), 165–172. Google Scholar

[3] P.K. Banerji, S.K. Al-Omari and L. Debnath, Tempered distributional sine (co-sine) transform, Integral Transforms Spec. Funct. 17 (11) (2006), 759–768. Google Scholar

[4] H.H. Bang, A property of infinitely differentiable functions, Proc. Amer. Math. Soc. 108 (1) (1990), 73–76. Google Scholar

[5] G. Birkoff and S. MacLane, A Survey of Modern Algebra, MacMillan, New York, 1965. Google Scholar

[6] C. Chettaoui, Y. Othmani and K. Trim`eche, On the range of the Dunkl transform on Rd, Anal. and Appl. 2 (3) (2004), 177–192. Google Scholar

[7] C. F. Dunkl, Differential-difference operators associated to reflection groups, Trans. Am. Math. Soc. 311 (1989), 167–183. Google Scholar

[8] C.F. Dunkl, Integral kernels with reflection group invariance, Can. J. Math. 43 (1991), 1213-1227. Google Scholar

[9] C.F. Dunkl, Hankel transforms associated to finite reflection groups, Contemp. Math. 138 (1992), 123–138. Google Scholar

[10] L. Gallardo and L. Godefroy, Liouville’s property and Poisson’s equation for the generalized Dunkl Laplacian, C. R. Math. Acad. Sci. Paris, 337 (10) (2003), 1-6. Google Scholar

[11] M.F.E. de Jeu, The Dunkl transform, Invent.Math. 113 (1993), 147–162. Google Scholar

[12] J.-P. Gabardo, Tempered distributions with spectral gaps, Math. Proc. Camb. Phil. Soc. 106 (1989), 143–162. Google Scholar

[13] R. Howard and M. Reese, Characterization of eigenfunctions by boundedness conditions, Canad. Math. Bull. 35 (1992), 204–213. Google Scholar

[14] H. Mejjaoli and K. Trim`eche, Spectrum of functions for the Dunkl transform on Rd, Fract. Calc. Appl. Anal. 10 (1) (2007), 19–38. Google Scholar

[15] H. Mejjaoli and R. Daher, Roe’s theorem associated with the Dunkl operators, Int. J. Mod. Math 5 (2010), 299–314. Google Scholar

[16] H. Mejjaoli and K. Trim`eche, Characterization of the support for the Hyper-geometric Fourier transform of the W-invariant functions and distributions on Rd and Roe’s theorem, Journal of inequalities and Applications, (2014):99 doi:10.1186/1029-242X-2014-99. Google Scholar

[17] H. Mejjaoli, Spectral theorems associated with the Dunkl type operator on the real line, Int. J. Open Problems Complex Analysis, 7 (2) June (2015), 17–42. Google Scholar

[18] H. Mejjaoli, Paley-Wiener theorems of generalized Fourier transform associated with a Cherednik type operator on the real line, Complex Anal. Oper. Theory, 10 (6) (2016), 1145–1170. Google Scholar

[19] J. Roe, A characterization of the sine function, Math. Proc. Comb. Phil. Soc. 87 (1980), 69–73. Google Scholar

[20] M.R ̈osler, HermitepolynomialsandtheheatequationforDunkloperators, Comm. Math. Phys. 192 (1998), 519–542. Google Scholar

[21] R.S. Strichartz, Characterization of eigenfunctions of the Laplacian by bounded- ness conditions, Trans. Amer. Math. Soc. 338 (1993), 971–979. Google Scholar

[22] S. Thangavelu and Y. Xu, Convolution operator and maximal functions for Dunkl transform, J. d’Analyse Mathematique 97 (2005), 25–56. Google Scholar

[23] V.K. Tuan, Paley-Wiener theorems for a class of integral transforms, J. Math. Anal. Appl. 266 (2002), 200–226. Google Scholar

[24] K. Trim`eche, Paley-Wiener theorems for Dunkl transform and Dunkl translation operators, Integ. Transf. and Special Funct. 13 (2002), 17–38. Google Scholar