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Huygens’ principle for the wave equation associated with the trigonometric Dunkl-Cherednik operators. (English) Zbl 1088.39018
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.

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
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