# The Analysis of Linear Partial Differential Operators I

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Juli 2003

## Beschreibung

### Beschreibung

The main change in this edition is the inclusion of exercises with answers and hints. This is meant to emphasize that this volume has been written as a general course in modern analysis on a graduate student level and not only as the beginning of a specialized course in partial differen tial equations. In particular, it could also serve as an introduction to harmonic analysis. Exercises are given primarily to the sections of gen eral interest; there are none to the last two chapters. Most of the exercises are just routine problems meant to give some familiarity with standard use of the tools introduced in the text. Others are extensions of the theory presented there. As a rule rather complete though brief solutions are then given in the answers and hints. To a large extent the exercises have been taken over from courses or examinations given by Anders Melin or myself at the University of Lund. I am grateful to Anders Melin for letting me use the problems originating from him and for numerous valuable comments on this collection. As in the revised printing of Volume II, a number of minor flaws have also been corrected in this edition. Many of these have been called to my attention by the Russian translators of the first edition, and I wish to thank them for our excellent collaboration.### Inhaltsverzeichnis

I. Test Functions.- Summary.

- 1.1. A review of Differential Calculus.

- 1.2. Existence of Test Functions.

- 1.3. Convolution.

- 1.4. Cutoff Functions and Partitions of Unity.- Notes.-

II. Definition and Basic Properties of Distributions.- Summary.

- 2.1. Basic Definitions.

- 2.2. Localization.

- 2.3. Distributions with Compact Support.- Notes.-

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.-

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.-

V. Distributions in Product Spaces.- Summary.

- 5.1. Tensor Products.

- 5.2. The Kernel Theorem.- Notes.-

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.-

VII. The Fourier Transformation.- Summary.

- 7.1. The Fourier Transformation in ? and in ?'.

- 7.2. Poisson's Summation Formula and Periodic Distributions.

- 7.3. The Fourier-Laplace Transformation in ?'.

- 7.4. More General Fourier-Laplace Transforms.

- 7.5. The Malgrange Preparation Theorem.

- 7.6. Fourier Transforms of Gaussian Functions.

- 7.7. The Method of Stationary Phase.

- 7.8. Oscillatory Integrals.

- 7.9. H(s), Lp and Hölder Estimates.- Notes.-

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 of 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.-

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.- Index of Notation.

### Portrait

Biography of Lars HörmanderBorn on January 24, 1931, on the southern coast of Sweden, Lars Hörmander did his secondary schooling as well as his undergraduate and doctoral studies in Lund. His principle teacher and adviser at the University of Lund was Marcel Riesz until he returned, then Lars Gårding. In 1956 he worked in the USA, at the universities of Chicago, Kansas, Minnesota and New York, before returning to a chair at the University of Stockholm. He remained a frequent visitor to the US, particularly to Stanford and was Professor at the IAS, Princeton from 1964 to 1968. In 1968 he accepted a chair at the University of Lund, Sweden, where, today, he is Emeritus Professor.

Hörmanders lifetime work has been devoted to the study of partial differential equations and its applications in complex analysis. In 1962 he was awarded the Fields Medal for his contributions to the general theory of linear partial differential operators. His book Linear Partial Differential Operators published 1963 by Springer in the Grundlehren series was the first major account of this theory. Hid four volume text The Analysis of Linear Partial Differential Operators published in the same series 20 years later illustrates the vast expansion of the subject in that period.

EAN: 9783540006626

ISBN: 3540006621

Untertitel: Distribution Theory and Fourier Analysis.
2nd ed. 2003.
Book.
Sprache: Englisch.

Verlag: Springer

Erscheinungsdatum: Juli 2003

Seitenanzahl: 456 Seiten

Format: kartoniert

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