Ben Saïd, Salem Huygens’ principle for the wave equation associated with the trigonometric Dunkl-Cherednik operators. (English) Zbl 1088.39018 Math. Res. Lett. 13, No. 1, 43-58 (2006). Summary: Let \({\mathfrak a}\) be an Euclidean vector space of dimension \(N\), and let \(k= (k_\alpha)_{\alpha\in{\mathcal R}}\) be a multiplicity function related to a root system \({\mathcal R}\). Let \(\Delta(k)\) be the trigonometric Dunkl-Cherednik differential-difference Laplacian [cf. I. Cherednik, Invent. Math. 106, No. 2, 411–432 (1991; Zbl 0725.20012)]. For \((a,t)\in \exp({\mathfrak a})\times\mathbb R\), denote by \(u_k(a,t)\) the solution to the wave equation \(\Delta(k) u_k(a,t)= \partial_{tt}u_k(a,t)\), where the initial data are supported inside a ball of radius \(R\) about the origin. We prove that \(u_k\) has support in the shell \(\{(a,t)\in \exp({\mathfrak a})\times\mathbb R\mid |t|-R\leq \|\log a\|\leq |t|+R\}\) if and only if the root system \({\mathcal R}\) is reduced, \(k_\alpha\in \mathbb N\) for all \(\alpha\in{\mathcal R}\), and \(N\) is odd starting from 3. Cited in 4 Documents MSC: 39A70 Difference operators 43A30 Fourier and Fourier-Stieltjes transforms on nonabelian groups and on semigroups, etc. 58J45 Hyperbolic equations on manifolds 35L05 Wave equation 35R10 Partial functional-differential equations Keywords:Dunkl-Cherednik operators; Huygens’ principle; Paley-Wiener theorem; wave equation Citations:Zbl 0725.20012 PDFBibTeX XMLCite \textit{S. Ben Saïd}, Math. Res. Lett. 13, No. 1, 43--58 (2006; Zbl 1088.39018) Full Text: DOI