## Abstract

Let k = (k_{α})_{αεℝ}, be a positive-real valued multiplicity function related to a root system ℝ, and Δ_{k} be the Dunkl-Laplacian operator. For (x, t) ε ℝ^{N}, × ℝ, denote by u_{k}(x, t) the solution to the deformed wave equation Δ_{k}u_{k},(x, t) = δ_{tt}u_{k}(x, t), where the initial data belong to the Schwartz space on ℝ^{N}. We prove that for k {greater than or slanted equal to} 0 and N {greater than or slanted equal to} l, the wave equation satisfies a weak Huygens' principle, while a strict Huygens' principle holds if and only if (N - 3)/2 + Σ_{αεℝ}+k_{α} ε ℕ. Here ℝ^{+} ⊂ ℝ is a subsystem of positive roots. As a particular case, if the initial data are supported in a closed ball of radius R > 0 about the origin, the strict Huygens principle implies that the support of u_{k}(x, t) is contained in the conical shell {(x, t), ε ℝ^{N} × ℝ | |t| - R {less-than or slanted equal to} {norm of matrix}x{norm of matrix} {less-than or slanted equal to} |t| + R}. Our approach uses the representation theory of the group SL(2, ℝ), and Paley-Wiener theory for the Dunkl transform. Also, we show that the (t-independent) energy functional of u_{k} is, for large |t|, partitioned into equal potential and kinetic parts.

Original language | English |
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Pages (from-to) | 351-391 |

Number of pages | 41 |

Journal | Indagationes Mathematicae |

Volume | 16 |

Issue number | 3-4 |

DOIs | |

Publication status | Published - Dec 19 2005 |

Externally published | Yes |

## Keywords

- Dunkl operators
- Huygens' principle
- Wave equation
- sl(2)-Triple

## ASJC Scopus subject areas

- General Mathematics