. Let X = G=K be an odd-dimensional semisimple Riemannian symmetric space of the noncompact type, and suppose that all Cartan subgroups of G are conjugate. Let u be a real-valued classical solution of the modified wave equation u tt = (\Delta + k)u on R \Theta X, the Cauchy data of which are supported in a closed metric ball of radius a at time t = 0. Here t is the coordinate on R, \Delta is the (nonpositive definite) Laplace-Beltrami operator on X, and k is a positive constant depending on the root structure of the Lie algebra of G. We show that the (t-independent) energy functional of u is eventually (for jtj a) partitioned into equal potential and kinetic parts; specifically, half the integrals over X of u 2 t and jduj 2 \Gamma ku 2 respectively, where d is the exterior derivative in X. The proof uses Helgason's Paley-Wiener Theorem for X, the classical Paley-Wiener Theorem, and properties of Harish-Chandra's c function. 1. Introduction and statement of the theorem. The ce...

Dedicated to Gerrit van Dijk on the occasion of his 65th birthday Abstract. We prove a Paley-Wiener Theorem for a class of symmetric spaces of the compact type, in which all root multiplicities are even. This theorem characterizes functions of small support in terms of holomorphic extendability and exponential type of their (discrete) Fourier transforms. We also provide three independent new proofs of the strong Huygens ’ principle for a suitable constant shift of the wave equation on odd-dimensional spaces from our class.

The purpose of this article is to define the radiation fields on asymptotically hyperbolic manifolds and to use them to study scattering theory. The radiation fields on R n and on asymptotically Euclidean manifolds were introduced by F.G. Friedlander in a series of papers starting in the early 1960’s [10, 11, 12, 13, 14]. His program of using the radiation fields to obtain the scattering matrix in that general setting was

. We consider the property of vanishing logarithmic term (VLT) for the fundamental solution of the shifted Laplace-d'Alembert operators + b (b a constant), on pseudoRiemannian reductive symmetric spaces M . Our main result is that such an operator on the c-dual or Flensted-Jensen dual of M has the VLT property if and only if a corresponding operator on M does. For Lorentzian spaces, where the + b are hyperbolic, VLT is known to be equivalent to the strong Huygens principle. We use our results to construct a large supply of new (space, operator) pairs satisfying Huygens' principle. 1. Introduction and principal results. An old theme in geometry is that of partial differential equations on manifolds which have particularly nice fundamental solutions. For hyperbolic equations, one might want sharp, i.e. dispersion free, propagation of sharp signals. This property, Huygens' principle, can be detected by studying the asymptotics of the fundamental solution, in the vanishing of certain ter...

In this paper we give a review on normally hyperbolic operators of Huygens type. The methods to determine Huygens operators we explain here were essentially influenced and developed by Paul Gunther. Contents 1 Introduction 2 2 Riesz distributions 4 2.1 Riesz distributions on Minkowski spaces : : : : : : : : : : : : : : : : : : 4 2.2 Riesz distributions on geodesically normal domains in Lorentzian manifolds : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 6 3 Normally hyperbolic operators 8 3.1 Definition and examples : : : : : : : : : : : : : : : : : : : : : : : : : : : 8 3.2 The Weitzenbock formula : : : : : : : : : : : : : : : : : : : : : : : : : : 9 3.3 Hadamard coefficients : : : : : : : : : : : : : : : : : : : : : : : : : : : : 10 3.4 Fundamental solution : : : : : : : : : : : : : : : : : : : : : : : : : : : : 12 4 Huygens operators 14 5 Conformal gauge invariance of Huygens operators 15 6 The moments of normally hyperbolic operators 18 6.1 Mappings with simple tran...