Introduction.
Chapter I. Test Functions
Summary
A review of Differential Calculus
Existence of Test Functions
Convolution
Cutoff Functions and Partitions of Unity
Notes
Chapter II. Definition and Basic Properties of Distributions
Summary
2.1. Basic Definitions
2.2. Localizations
2.3. Distributions with Compact Support
Notes
Chapter III. Differentiation and Multiplication by Functions
Summary
3.1. Definition and Examples
3.2. Homogeneous Distributions
3.3. Some Fundamental Solutions
3.4. Evaluation of Some Integrals
Notes
Chapter IV. Convolution
Summary
4.1. Convolution with a Smooth Function
4.2. Convolution of Distributions
4.3. The Theorem of Supports
4.4. The Role of Fundamental Solutions
4.5. Basic Lp Estimates for Convolutions
Notes
Chapter V. Distributions in Product Spaces
Summary
5.1. Tensor Products
5.2. The Kernel Theorem
Notes
Chapter VI. Composition with Smooth Maps
Summary
6.1. Definitions
6.2. Some Fundamental Solutions
6.3. Distributions on a Manifold
6.4. The Tangent and Cotangent Bundles
Notes
Chapter VII. The Fourier Transformation
Summary
7.1. The Fourier Transformation in J and J ’
7.2. Piosson’s Summation Formula and Periodic Distributions
7.3. The Fourier-Laplace Transformation in e ’
7.4. More General Fourier-Laplace Transforms
7.5. The Malgrange Preparation Theorem
7.6. Fourier Transforms of Gaussian Functions
7.7. The Mthod of Stationary Phase
7.8. Oscillatory Integrals
7.9. H(8), Lp and Hölder Estimates
Notes
Chapter VIII. Spectral Analysis of Singularities
Summary
8.1.The Wave Front Set
8.2. A Review of Operations with Distributions
8.3. The Wave Front Set and Solutions of Partial Differential Equations
8.4. The Wave Front set with Respect to CL
8.5. Rules of Computation for WFL
8.6. WFL for Solutions of Partial Differential Equations
8.7. Microhyperbolicity
Notes
Chapter IX. Hyperfunctions
Summary
9.1. Analytic Functionals
9.2. General Hyperfunctions
9.3. The Analytic Wave Front Set of a Hyperfunction
9.4. The Analytic Cauchy Problem
9.5. Hyperfunction Solutions of Partial Differential Equations
9.6. The Analytic Wave Front Set and the Support
Notes
Exercises
Answers and Hints to All the Exercises
Bibliography
Index
Index of Notation