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## Über dieses Buch

During the last ten years a powerful technique for the study of partial differential equations with regular singularities has developed using the theory of hyperfunctions. The technique has had several important applications in harmonic analysis for symmetric spaces.

This book gives an introductory exposition of the theory of hyperfunctions and regular singularities, and on this basis it treats two major applications to harmonic analysis. The first is to the proof of Helgason’s conjecture, due to Kashiwara et al., which represents eigenfunctions on Riemannian symmetric spaces as Poisson integrals of their hyperfunction boundary values.

A generalization of this result involving the full boundary of the space is also given. The second topic is the construction of discrete series for semisimple symmetric spaces, with an unpublished proof, due to Oshima, of a conjecture of Flensted-Jensen.

This first English introduction to hyperfunctions brings readers to the forefront of research in the theory of harmonic analysis on symmetric spaces. A substantial bibliography is also included. This volume is based on a paper which was awarded the 1983 University of Copenhagen Gold Medal Prize.

## Inhaltsverzeichnis

### 1. Hyperfunctions and microlocal analysis — an introduction

Abstract
The theory of hyperfunctions is in some sense a generalization of the theory of distributions, as developed by L. Schwartz. On a compact real analytic manifold M, a hyperfunction is the same as an analytic functional, that is, a continuous linear functional on the space.A(M) of analytic functions on M (equipped with a certain topology of inductive limits). On noncompact real analytic manifolds hyperfunctions are most conveniently studied by cohomological methods introduced by M. Sato. In the first four sections of this chapter we develop this theory, introducing along the way some elementary sheaf theory. The last two sections consist of a brief introduction to microlocal analysis.
Henrik Schlichtkrull

### 2. Differential equations with regular singularities

Abstract
The theory of ordinary differential equations with regular singularities has been well established for many years (cf. Hilb[a]). Since many of the phenomena that occur in the general theory for partial differential equations already appears in the much simpler special case of ordinary equations, we will start with a short summary of this theory.
Henrik Schlichtkrull

### 3. Riemannian symmetric spaces and invariant differential operators — preliminaries

Abstract
In this chapter we give a short summary of some notation and well known results which we need in the sequel. The material can be found, for instance, in Helgason’s books [j] and [n], except for the results of Section 3.2, where we refer to Varadarajan [b].
Henrik Schlichtkrull

### 4. A compact imbedding

Abstract
Let X be a Riemannian symmetric space. It is the purpose of this chapter to construct an imbedding of X into a compact real analytic manifold X̃. For the study of the asymptotic behavior of functions on X, which we shall carry out in the next chapters, this is of crucial importance.
Henrik Schlichtkrull

### 5. Boundary values and Poisson integral representations

Abstract
Consider the open disk D = {|z| > 1} in C with the boundary T = {|z| = 1}. The classical Poisson kernel is defined by
$$P\left( {z,t} \right) = \tfrac{{1 - {{\left| z \right|}^2}}}{{{{\left| {t - z} \right|}^2}}}$$
(5.1)
for z D , t T , and the Poisson transform Pf on D of a function f on T is given by
$$Pf(z) = \int {_Tf(t)\;P} \left( {z,t} \right)dt$$
(5.2)
Henrik Schlichtkrull

### 6. Boundary values on the full boundary

Abstract
In the preceding chapter we have represented the joint eigenfunctions on G/K as Poisson integrals of their hyperfunction boundary values on K/M. When G/K has rank < 1 this is, however, only a small part of the boundary of G/K in X̃, and it is important to have analogous results for the other G-orbits in the boundary. In this chapter we therefore generalize the results of Chapter 5 to this situation.
Henrik Schlichtkrull

### 7. Semisimple symmetric spaces

Abstract
In this chapter some basic properties of semisimple symmetric spaces are established, and a fundamental family of functions related to a semisimple symmetric space is defined. In the next chapter we shall use these results to study harmonic analysis on semisimple symmetric spaces.
Henrik Schlichtkrull

### 8. Construction of functions with integrable square

Abstract
Let G/H be a semisimple symmetric space. Since H is reductive it follows from Helgason [n] Chapter 1, Theorem 1.9 that G/H has an invariant measure, unique up to scalars. Hence the Hilbert space L2 (G/H) makes sense, and we can study the unitary representation
$$\left( {\pi (g)f} \right)\left( {x\;H} \right) = f\left( {{g^{ - 1}}x\;H} \right)$$
(g,x G) of G on this space. It is the purpose of L2-harmonic analysis on G/H to give an explicit decomposition (in general as a direct integral) of this representation into irreducibles. So far this program has not been accomplished in general (although the 2 answer is known in several specific cases, notably those of L2 (G/K) and L2 (G × G/d(g)) ≅ L2(G), by the work of Harish-Chandra — see the notes at the end of this chapter).
Henrik Schlichtkrull

### Backmatter

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