The Analysis of Linear Partial Differential Operators I
Subtitle: Distribution Theory and Fourier Analysis
By: by Lars Hörmander
Series: Grundlehren der mathematischen Wissenschaften, A Series of Comprehensive Studies in Mathematics -- 256; Grundlehren der mathematischen Wissenschaften, A Series of Comprehensive Studies in Mathematics -- 256.
Publisher: Springer Berlin Heidelberg
Published: 1998
Place of publication: Berlin, Heidelberg
Language: eng
ISBN-13: 9783642967504; 9783642967528
Subjects: Mathematics; Topological Groups; Lie Groups Topological Groups
Description: In 1963 my book entitled "Linear partial differential operators" was published in the Grundlehren series. Some parts of it have aged well but others have been made obsolete for quite some time by techniques using pseudo-differential and Fourier integral operators. The rapid de­ velopment has made it difficult to bring the book up to date. Howev­ er, the new methods seem to have matured enough now to make an attempt worth while. The progress in the theory of linear partial differential equations during the past 30 years owes much to the theory of distributions created by Laurent Schwartz at the end of the 1940's. It summed up a great deal of the experience accumulated in the study of partial differ­ ential equations up to that time, and it has provided an ideal frame­ work for later developments. "Linear partial differential operators" be­ gan with a brief summary of distribution theory for this was still un­ familiar to many analysts 20 years ago. The presentation then pro­ ceeded directly to the most general results available on partial differ­ ential operators. Thus the reader was expected to have some prior fa­ miliarity with the classical theory although it was not appealed to ex­ plicitly. Today it may no longer be necessary to include basic distribu­ tion theory but it does not seem reasonable to assume a classical background in the theory of partial differential equations since mod­ ern treatments are rare.
Table of contents: title: I. Test Functions; title: Summary; title: 1.1. A review of Differential Calculus; title: 1.2. Existence of Test Functions; title: 1.3. Convolution; title: 1.4. Cutoff Functions and Partitions of Unity; title: Notes; title: II. Definition and Basic Properties of Distributions; title: 2.1. Basic Definitions; title: 2.2. Localization; title: 2.3. Distributions with Compact Support; title: III. Differentiation and Multiplication by Functions; title: 3.1. Definition and Examples; title: 3.2. Homogeneous Distributions; title: 3.3. Some Fundamental Solutions; title: 3.4. Evaluation of Some Integrals; title: IV. Convolution; title: 4.1. Convolution with a Smooth Function; title: 4.2. Convolution of Distributions; title: 4.3. The Theorem of Supports; title: 4.4. The Role of Fundamental Solutions; title: 4.5. Basic Lp Estimates for Convolutions; title: V. Distributions in Product Spaces; title: 5.1. Tensor Products; title: 5.2. The Kernel Theorem; title: VI. Composition with Smooth Maps; title: 6.1. Definitions; title: 6.2. Some Fundamental Solutions; title: 6.3. Distributions on a Manifold; title: 6.4. The Tangent and Cotangent Bundles; title: VII. The Fourier Transformation; title: 7.1. The Fourier Transformation in $\cal S$ and in $\cal S$’,; title: 7.2. Poisson’s Summation Formula and Periodic Distributions; title: 7.3. The Fourier-Laplace Transformation in ?’,; title: 7.4. More General Fourier-Laplace Transforms; title: 7.5. The Malgrange Preparation Theorem; title: 7.6. Fourier Transforms of Gaussian Functions; title: 7.7. The Method of Stationary Phase; title: 7.8. Oscillatory Integrals; title: 7.9. H(s), Lp and Hölder Estimates; title: VIII. Spectral Analysis of Singularities; title: 8.1. The Wave Front Set; title: 8.2. A Review of Operations with Distributions; title: 8.3. The Wave Front Set of Solutions of Partial Differential Equations; title: 8.4. The Wave Front Set with Respect to CL; title: 8.5. Rules of Computation for WFL; title: 8.6. WFL for Solutions of Partial Differential Equations; title: 8.7. Microhyperbolicity; title: IX Hyperfunctions; title: 9.1. Analytic Functionals; title: 9.2. General Hyperfunctions; title: 9.3. The Analytic Wave Front Set of a Hyperfunction; title: 9.4. The Analytic Cauchy Problem; title: 9.5. Hyperfunction Solutions of Partial Differential Equations; title: 9.6. The Analytic Wave Front Set and the Support; title: Index of Notation.