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

Lars Hormander9780387523453, 0387523456

Due to popular demand this classic presentation of a vast amount on linear partial differential equations by a consummate master of the subject is now available as a study edition. The main change in this new edition is the inclusion of exercises with answers and hints. That is meant to emphasize that this volume can perfectly serve as a general course in modern analysis on a graduate student level and not only as a beginning of a specialised course in partial differential equations. In particular, it could also serve as an introduction to harmonic analysis. Exercises are given primarily to the sections of general interest. As in the revised printing of volume II, a number of minor flaws have also been corrected in this edition. Parallely this edition is still available as volume 256 of the Grundlehren der mathematischen Wissenschaften. “… it is the best now available in print. … All the theorems are there (among them the Schwartz kernel theorem), and all they have … proofs.” Bulletin of the American Mathematical Society#1 “It certainly will be a classic for many years.” Zentralblatt für Mathematik#2

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Table of contents :
Title page……Page 1
Series……Page 2
Title……Page 3
Date-line……Page 4
Preface to the Second Edition……Page 5
Preface……Page 7
Contents……Page 9
Introduction……Page 13
1.1. A review of Differential Calculus……Page 17
1.2. Existence of Test Functions……Page 26
1.3. Convolution……Page 28
1.4. Cutoff Functions and Partitions of Unity……Page 37
Notes……Page 43
2.1. Basic Definitions……Page 45
2.2. Localization……Page 53
2.3. Distributions with Compact Support……Page 56
Notes……Page 64
3.1. Definition and Examples……Page 66
3.2. Homogeneous Distributions……Page 80
3.3. Some Fundamental Solutions……Page 91
3.4. Evaluation of Some Integrals……Page 96
Notes……Page 98
Summary……Page 99
4.1. Convolution with a Smooth Function……Page 100
4.2. Convolution of Distributions……Page 112
4.3. The Theorem of Supports……Page 117
4.4. The Role of Fundamental Solutions……Page 121
4.5. Basic $L^p$ Estimates for Convolutions……Page 128
Notes……Page 136
5.1. Tensor Products……Page 138
5.2. The Kernel Theorem……Page 140
Notes……Page 144
6.1. Definitions……Page 145
6.2. Some Fundamental Solutions……Page 149
6.3. Distributions on a Manifold……Page 154
6.4. The Tangent and Cotangent Bundles……Page 158
Notes……Page 168
Summary……Page 170
7.1. The Fourier Transformation in $mathcal{J}$ and in $mathcal{J}’$……Page 171
7.2. Poisson’s Summation Formula and Periodic Distributions……Page 189
7.3. The Fourier-Laplace Transformation in $mathcal{E}’$……Page 193
7.4. More General Fourier-Laplace Transforms……Page 203
7.5. The Malgrange Preparation Theorem……Page 207
7.6. Fourier Transforms of Gaussian Functions……Page 217
7.7. The Method of Stationary Phase……Page 227
7.8. Oscillatory Integrals……Page 248
7.9. $H_{(s)}$, $L^p$ and Hoelder Estimates……Page 252
Notes……Page 260
Summary……Page 263
8.1. The Wave Front Set……Page 264
8.2. A Review of Operations with Distributions……Page 273
8.3. The Wave Front Set of Solutions of Partial Differential Equations……Page 283
8.4. The Wave Front Set with Respect to $C^L$……Page 292
8.5. Rules of Computation for $WF_L$……Page 308
8.6. $WF_L$ for Solutions of Partial Differential Equations……Page 317
8.7. Microhyperbolicity……Page 329
Notes……Page 334
Summary……Page 337
9.1. Analytic Functionals……Page 338
9.2. General Hyperfunctions……Page 347
9.3. The Analytic Wave Front Set of a Hyperfunction……Page 350
9.4. The Analytic Cauchy Problem……Page 358
9.5. Hyperfunction Solutions of Partial Differential Equations……Page 365
9.6. The Analytic Wave Front Set and the Support……Page 370
Notes……Page 380
Exercises……Page 383
Answers and Hints to All the Exercises……Page 406
Bibliography……Page 431
Index……Page 449
Index of Notation……Page 451