Huygens’ principle and equipartition of energy for the modified wave equation associated to a generalized radial Laplacian

Abstract

In this paper we consider the modified wave equation associated with a class of radial Laplacians L generalizing the radial part of the Laplace-Beltrami operator on hyperbolic spaces or Damek-Ricci spaces. We show that the Huygens' principle and the equipartition of energy hold if the inverse of the Harish-Chandra c-function is a polynomial and that these two properties hold asymptotically otherwise. Similar results were established previously by Branson, Olafsson and Schlichtkrull in the case of noncompact symmetric spaces.

The Shifted Wave Equation on Damek–Ricci Spaces and on Homogeneous Trees

“…In even dimension, u(x, t) may not vanish when d(x, x 0 ) < |t|−R, but it tends asymptotically to 0. This phenomenon was observed in several settings, for instance on Euclidean spaces in [27], on Riemannian symmetric spaces of the noncompact type [7], on Damek-Ricci spaces [4], for Chébli-Trimèche hypergroups [14], ... Our next result differs from [7,14,4] in two ways. On one hand, we use explicit expressions instead of the Fourier transform.…”

“…the potential energy. By the way, let us mention that the equipartition of (11) into kinetic and potential energies was investigated in [7] and in the subsequent works [14,5,4] (see also [18,Section V.5.5] and the references cited therein).…”

“…Let us mention [7] (see also [18,Section V.5]) for Huygens' principle and the energy equipartition on Riemannian symmetric spaces of the noncompact type. This work was extended to Damek-Ricci spaces in [4], to Chébli-Trimèche hypergroups in [14] and to the trigonometric Dunkl setting in [6,5]. The nonlinear shifted wave equation was studied in [28,2,3], first on real hyperbolic spaces and next on Damek-Ricci spaces.…”

Huygens’ principle and a Paley–Wiener type theorem on Damek–Ricci spaces

“…We also would like to mention that Huygens' principle for the radial part of the Laplace-Beltrami operator on Damek-Ricci spaces has been studied by J. El Kamel and C. Yacoub [14] and by Branson, Ólafsson and A. Pasquale [8] in the general context of Jacobi operators. Moreover, the case of the Dunkl-Cherednik Laplacian has been studied by F. Ayadi [6] and S. Ben Saïd [7].…”