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Paley-Wiener theorems on a symmetric space and their application. (English) Zbl 0786.43005

The principal aim of this paper is to study the Fourier analysis on Riemannian symmetric spaces. In particular, the Paley-Wiener theorems for some types of function spaces on a symmetric space will be discussed throughout the paper. As an application of these results, the solvability of a single differential equation defined by an invariant differential operator on a symmetric space will be shown in the last section. Furthermore, in the forthcoming paper we will discuss Ehrenpreis’ fundamental principle on a Riemannian symmetric space. The statement of this fundamental principle was given by the authors [in Algebraic analysis 2, 681-698 (1989; Zbl 0705.47040)]. The present paper is a preliminary investigation for the proof of this fundamental principle.
In Section 1 we briefly set up the notations and basic facts about real reductive linear Lie groups and define Riemannian symmetric spaces we deal with throughout the paper. In Section 2 we introduce the notion of the invariant differential operator of infra-exponential type which is, in fact, an infinite order differential operator on our symmetric space. Using these differential operators we define the function space \({\mathcal A}_ *(G/K)\). Next we define the function space \({\mathcal C}_ *(G/K)\). This space is the inductive limit of the spaces of \(L^ p\) Schwartz functions discussed by M. Eguchi [J. Funct. Anal. 34, 167-210 (1979; Zbl 0433.43012)] and has the structure of the FS space. The space \({\mathcal A}_ *(G/K)\) is the subspace of \({\mathcal C}_ *(G/K)\) consisting of those real analytic functions which belong to \({\mathcal C}_ *(G/K)\). In Section 3 we review the Fourier-Laplace transformations on symmetric spaces and state the main results (Theorem 1). We prove the theorem for the case of \({\mathcal C}_ *(G/K)\) in Section 4. Since the proof is based on the results of [loc. cit.], we review them in the same section. In order to prove the theorem in the case of \({\mathcal A}_ *(G/K)\), we must get some knowledge of the Fourier series of analytic functions on the isotropy group at the origin of the symmetric space. We discuss them in detail in Section 5. Using the theorem in the case of \({\mathcal C}_ *(G/K)\) and the results of Section 5, we get the proof of the theorem in the case of \({\mathcal A}_ *(G/K)\) in Section 6. For use in a forthcoming paper, we describe the Fourier coefficients of the Fourier-Laplace images of our function spaces on the boundary of the symmetric space. In the last section we derive some results on the solvability of a single differential equation on a symmetric space.
In Appendix 1, we prove some lemmas fundamental for our discussion of the differential operators of infra-exponential type. Elementary lemmas of the infra-exponential functions and the important properties of the differential operators of infra-exponential type deduced from the results of Appendix 1 are collected in Appendix 2.

MSC:

43A85 Harmonic analysis on homogeneous spaces
53C35 Differential geometry of symmetric spaces
47F05 General theory of partial differential operators
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