×

\(L^{p}\)-Fourier multipliers for the Dunkl operator on the real line. (English) Zbl 1045.43003

The Dunkl transform \({\mathcal F}_\alpha\), \(\alpha\geq -1/2\), is a generalization of the Fourier transform \({\mathcal F}\), which corresponds to \(\alpha=-1/2\). Starting with the Dunkl operator: \[ (\Lambda_\alpha f)(x)={d\over dx}\,f(x) + {2\alpha+1\over x}\bigg[{f(x)-f(-x)\over 2}\bigg], \] one considers the Dunkl kernel \(E_\alpha(\lambda x)\), \(\lambda\in {\mathbb C}\,\), which is the unique solution of the equation \(\Lambda_\alpha f(x)=\lambda f(x)\) with \(f(0)=1\). An explicit expression of \(E_\alpha(\lambda x)\) can be given, using a series with the gamma function, and \(E_{-1/2}(\lambda x)=\text{ e}^{\lambda x}\). The Dunkl transform is then defined, for \(\lambda\in {\mathbb R}\), by: \[ \big({\mathcal F}_\alpha f\big)(\lambda)= \int_{\mathbb R} E_\alpha (-i\lambda x) f(x)\,d\mu_\alpha(x), \] where \(d\mu_\alpha(x)= \big(2^{\alpha+1} \Gamma(\alpha+1)\big)^{-1}| x|^{2\alpha+1} dx\). An inversion formula and a Plancherel theorem are available for the Dunkl transform [M. F. E. de Jeu, Invent. Math. 113, No. 1, 147–162 (1993; Zbl 0789.33007)]. The aim of this paper is to prove the analogue of Hörmander’s theorems on Fourier multipliers for the Dunkl transform. For this, the author needs weighted Sobolev spaces. His results allow the author to give some examples of multipliers. In the last part, the author investigates the \(L^p-L^q\) boundedness of multipliers (\(q>p\)).
Reviewer: Daniel Li (Lens)

MSC:

43A15 \(L^p\)-spaces and other function spaces on groups, semigroups, etc.
42A38 Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type
42A45 Multipliers in one variable harmonic analysis
42B10 Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type
43A32 Other transforms and operators of Fourier type
44A15 Special integral transforms (Legendre, Hilbert, etc.)

Citations:

Zbl 0789.33007
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Anker, J. Ph., \(L^p\) Fourier multipliers on Riemannian symmetric spaces of the noncompact type, Ann. of Math., 132, 597-628 (1990) · Zbl 0741.43009
[2] Baker, T. H.; Forrester, P. J., The Calogero-Sutherland model and generalized classical polynomials, Comm. Math. Phys., 188, 175-216 (1997) · Zbl 0903.33010
[3] Bergh, J.; Löfström, J., Interpolation Spaces an Introduction. Interpolation Spaces an Introduction, Grundlehren Math. Wiss, Vol. 223 (1976), Springer: Springer Berlin, New York · Zbl 0344.46071
[4] Bloom, W. R.; Xu, Z., Fourier multipliers for \(L^p\) on Chébli-Trimèche hypergroups, Proc. London Math. Soc., 80, 643-664 (2000) · Zbl 1022.43005
[5] Cherednik, L., A unification of the Knizhnik-Zamolodchikov equation and Dunkl operators via affine Hecke algebras, Invent. Math., 106, 411-432 (1991) · Zbl 0725.20012
[6] Coifman, R. R.; Weiss, G., Analyse Harmonique Non-commutative sur Certains espaces Homogènes. Analyse Harmonique Non-commutative sur Certains espaces Homogènes, Lecture Notes in Mathematics, Vol. 242 (1971), Springer: Springer Berlin, New York · Zbl 0224.43006
[7] DeMichele, L.; Inglis, I. R., \(L^p\) estimates for strongly singular integrals on spaces of homogeneous type, J. Funct. Anal., 39, 1-15 (1980) · Zbl 0461.46039
[8] Dunkl, C. F., Differential-difference operators associated with reflections groups, Trans. Amer. Math. Soc., 311, 167-183 (1989) · Zbl 0652.33004
[9] Dunkl, C. F., Integral kernels with reflection group invariance, Canad. J. Math., 43, 1213-1227 (1991) · Zbl 0827.33010
[10] Edwards, R. E.; Gaudry, G. I., Littlewood-Paley and Multiplier Theory (1977), Springer: Springer Berlin, Heidelberg, New York · Zbl 0464.42013
[11] de Jeu, M. F.E., The Dunkl transform, Invent. Math., 113, 147-162 (1993) · Zbl 0789.33007
[12] Gosselin, J.; Stempak, K., A weak-type estimate for Fourier-Bessel multipliers, Proc. Amer. Math. Soc., 106, 622-655 (1989) · Zbl 0684.42007
[13] Hörmander, L., Estimates for translation invariant operators in \(L^p\) spaces, Acta Math., 104, 93-140 (1960) · Zbl 0093.11402
[14] Lapointe, L.; Vinet, L., Exact operator solution of the Calogero-Sutherland model, Comm. Math. Phys., 178, 425-452 (1996) · Zbl 0859.35103
[15] Opdam, E. M., Harmonic analysis for certain representations of graded Hecke algebras, Acta Math., 175, 75-121 (1995) · Zbl 0836.43017
[16] M. Rösler, Bessel-type signed hypergroups on \(R\); M. Rösler, Bessel-type signed hypergroups on \(R\)
[17] Sifi, M.; Soltani, F., Generalized Fock spaces and Weyl relations for the Dunkl kernel on the real line, J. Math. Anal. Appl., 270, 92-106 (2002) · Zbl 1012.46033
[18] Stein, E. M., Singular Integrals and Differentiability Properties of Functions (1970), Princeton University Press: Princeton University Press Princeton · Zbl 0207.13501
[19] Triebel, H., Interpolation Theory, Function Spaces, Differential Operators (1978), North-Holland: North-Holland Amsterdam, New York, Oxford · Zbl 0387.46032
[20] Trimèche, K., Paley-Wiener Theorems for the Dunkl transform and Dunkl translation operators, Internat. Trans. Spec. Funct., 13, 17-38 (2002) · Zbl 1030.44004
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.