Inversion Formula for the Dunkl-Wigner Transform and Compactness Property for the Dunkl-Weyl Transforms

Abstract:

      We define and study the Fourier-Wigner transform associated with the Dunkl operators, and we prove for this transform an inversion formula. Next, we introduce and study the Weyl transforms $W_{\sigma}$ associated with the Dunkl operators, where $\sigma$ is a symbol in the Schwartz space $\mathcal{S}(\mathbb{R}^d\times\mathbb{R}^d)$. An integral relation between the precedent Weyl and Wigner transforms is given. At last, we give criteria in terms of $\sigma$ for boundedness and compactness of the transform $W_{\sigma}$.