## $$L^{p}$$-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.)

Zbl 0789.33007
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