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 The second topic is the construction of discrete series for semisimple symmetric spaces, with an unpublished proof, due to Oshima, of a conjecture of Flensted-Jensen.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. - 1. 1. Hyperfunctions of one variable. - 1. 2. Sheaves. - 1. 3. Cohomology of sheaves. - 1. 4. Hyperfunctions of several variables. - 1. 5. The singular spectrum and microfunctions. - 1. 6. Micro-differential operators. - 1. 7. Notes. - 2. Differential Equations with Regular Singularities. - 2. 1. Regular singularities for ordinary equations. - 2. 2. Regular singularities for partial differential equations. - 2. 3. Boundary values for a single equation. - 2. 4. Example. - 2. 5. Boundary values for a system of equations. - 2. 6. Notes. - 3. Riemannian Symmetric Spaces and Invariant Differential Operators Preliminaries. - 3. 1. Decomposition and integral formulas for semisimple Lie groups. - 3. 2. Parabolic subgroups. - 3. 3. Invariant differential operators. - 3. 4. Notes. - 4. A Compact Imbedding. - 4. 1. Construction and analytic structure of X? . - 4. 2. Invariant differential operators on X? . - 4. 3. Regular singularities. - 4. 4. Notes. - 5. Boundary Values and Poisson Integral Representations. - 5. 1. Poisson transformations. - 5. 2. Boundary value maps. - 5. 3. Spherical functions and their asymptotics. - 5. 4. Integral representations. - 5. 5. Notes and further results. - 6. Boundary Values on the Full Boundary. - 6. 1. Partial Poisson transformations. - 6. 2. Partial spherical functions and Poisson kernels. - 6. 3. Boundary values and asymptotics. - 6. 4. The bijectivity of the partial Poisson transformations. - 6. 5. Notes and further results. - 7. Semisimple Symmetric Spaces. - 7. 1. The orbits of symmetric subgroups. - 7. 2. Root systems. - 7. 3. A fundamental family of functions. - 7. 4. A differential property. - 7. 5. Asymptotic expansions. - 7. 6. The case of equal rank. - 7. 7. Examples. - 7. 8. Notes and further results. - 8. Construction ff Functions with IntegrableSquare. - 8. 1. The invariant measure on G/H. - 8. 2. An important duality. - 8. 3. Discrete series. - 8. 4. Examples. - 8. 5. Notes and further results.
