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

Francesca Astengo,Bianca Di Blasio

doi:10.5802/ambp.286

Abstract

We prove that Huygens' principle and the principle of equipartition of energy hold for the modified wave equation on odd dimensional Damek-Ricci spaces. We also prove a Paley-Wiener type theorem for the inverse of the Helgason Fourier transform on Damek-Ricci spaces.

Highlights

On a Riemannian manifold X, consider the Cauchy problem for the modified wave equationL + c u = utt f, g ∈ Cc∞ (X), u(·, 0) = f (1.1)u (·, 0) = g t where L is the Laplace–Beltrami operator on X and c is a suitable constant
In this paper we give a simple proof of the converse for Damek–Ricci spaces and we show the exponential and strict version of the principle of equipartition of energy
Di Blasio of Damek–Ricci spaces a Paley–Wiener type theorem for the inverse Helgason Fourier transform proved by N

Summary

Introduction

On a Riemannian manifold X, consider the Cauchy problem for the modified wave equation. Schlichtkrull [9] studied Huygens’ principle and principle of equipartition of energy for the modified wave equation on noncompact symmetric spaces. We follow their approach applying the following result proved in [5, 4] (see Theorem 2.2 below): the Helgason Fourier transform of a smooth function with compact support on a. We 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. Noguchi describes the solution of the modified wave equation on Damek–Ricci spaces in terms of means over geodesic spheres and the heat kernel As an application, he shows that Huygens’ principle holds on odd dimensional Damek–Ricci spaces.

Notation and preliminaries

Huygens’ principle

Equipartition of energy

Paley–Wiener type Theorem