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The Analysis of Linear Partial Differential Operators I: Distribution Theory and Fourier Analysis

Lars Hörmander
Publisher: 
Springer
Publication Date: 
2003
Number of Pages: 
440
Format: 
Paperback
Series: 
Classics in Mathematics
Price: 
59.95
ISBN: 
978-3540006626
Category: 
Monograph
BLL Rating: 

The Basic Library List Committee recommends this book for acquisition by undergraduate mathematics libraries.

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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

Dummy View - NOT TO BE DELETED