Hyperfunctions and Harmonic Analysis on Symmetric Spaces

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 Integrable Square.- 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.