\(L^{p}\)-Fourier multipliers for the Dunkl operator on the real line.
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 [\textit{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 (\(q>p\)).
scientific article; zbMATH DE number 2073657
scientific article