Spectral theorems associated to the Dunkl operators

Hatem Mejjaoli

Abstract


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.

Keywords


Dunkl transform, Dunkl resolvent function, real Paley-Wiener theorem, Roe's theorem.

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References


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.

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

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.

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

G. Birkoff and S. MacLane, A Survey of Modern Algebra, MacMillan, New York, 1965.

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.

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

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

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

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.

M.F.E. de Jeu, The Dunkl transform, Invent.Math. 113 (1993), 147–162.

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

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

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

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

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.

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.

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.

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

M.R ̈osler, HermitepolynomialsandtheheatequationforDunkloperators, Comm. Math. Phys. 192 (1998), 519–542.

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

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

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

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




DOI: http://dx.doi.org/10.11568/kjm.2016.24.4.693

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