A Paley-Wiener theorem for the spherical Laplace transform on causal symmetric spaces of rank 1

Abstract

We formulate and prove a topological Paley-Wiener theorem for the normalized spherical Laplace transform defined on the rank 1 causal symmetric spaces M = SO0(1,n)/SO(1,n -1), for n > 2. INTRODUCTION The spherical Laplace transform on causal symmetric spaces was introduced in [FHO, ?8] as a generalization of the spherical Fourier transform on Riemannian symmetric spaces defined by Helgason (see [Hi, Chapter 4]). Both transforms can be expressed in terms of (integrating against) spherical functions. It was furthermore shown in [01, ?5] that the spherical functions on the Riemannian dual of a causal symmetric space can be written as an expansion in spherical functions on the causal symmetric space. The inversion formula for the spherical Laplace transform easily follows (see [01, ?6]). One of the most important results on the spherical Fourier transform is the (topological) Paley-Wiener theorem (see [Hi, Chapter 4, ?7] and [H2, Chapter 3, ?5] for details) generalizing the classical Paley-Wiener theorem on Euclidean spaces. In this paper we generalize these results to the normalized spherical Laplace transform on causal symmetric spaces M of rank 1, thereby partially solving an open problem posed by the second author in [02, ?5]. The paper is divided into two sections: in the first section we recall some results on the spherical Fourier transform on the Riemannian dual Md of M, and in the second we consider the spherical Laplace transform defined on M. We define the Paley-Wiener space, the supposed image space of spherical Laplace transform, according to the growth and symmetry conditions satisfied by the spherical functions on M. The Paley-Wiener theorem for the normalized spherical Laplace transform follows by using Cauchy's theorem to change the path of integration in the inversion formula and from the Paley-Wiener theorem for the spherical Fourier transform on Md. Received by the editors December 9, 1998 and, in revised form, March 22, 1999. 2000 Mathematics Subject Classification. Primary 43A85, 22E30; Secondary 43A90, 33C60. The first author was supported by a postdoc fellowship from the European Commission within the European TMR Network Harmonic Analysis 1998-2001 (Contract ERBFMRX-CT97-0159). The second author was supported by LEQSF grant (1996-99)-RD-A-12. (?2000 American Mathematical Society