The Analysis of Linear Partial Differential Operators I: Distribution Theory and Fourier Analysis (Grundlehren der mathematischen Wissenschaften) (v. 1)

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. 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 2.1. Basic Definitions 2.2. Localization 2.3. Distributions with Compact Support III. Differentiation and Multiplication by Functions 3.1. Definition and Examples 3.2. Homogeneous Distributions 3.3. Some Fundamental Solutions 3.4. Evaluation of Some Integrals IV. Convolution 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 V. Distributions in Product Spaces 5.1. Tensor Products 5.2. The Kernel Theorem VI. Composition with Smooth Maps 6.1. Definitions 6.2. Some Fundamental Solutions 6.3. Distributions on a Manifold 6.4. The Tangent and Cotangent Bundles VII. The Fourier Transformation 7.1. The Fourier Transformation in $\cal S$ and in $\cal S$’, 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 VIII. Spectral Analysis of Singularities 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 IX Hyperfunctions 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 Index of Notation. 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 fornumerous 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. 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 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 "It certainly will be a classic for many years." Zentralblatt f?r Mathematik Front Matter....Pages I-IX Introduction....Pages 1-4 Test Functions....Pages 5-32 Definition and Basic Properties of Distributions....Pages 33-53 Differentiation and Multiplication by Functions....Pages 54-86 Convolution....Pages 87-125 Distributions in Product Spaces....Pages 126-132 Composition with Smooth Maps....Pages 133-157 The Fourier Transformation....Pages 158-250 Spectral Analysis of Singularities....Pages 251-324 Hyperfunctions....Pages 325-370 Back Matter....Pages 371-394 Author received the 1962 Fields MedalAuthor received the 1988 Wolf Prize (honoring achievemnets of a lifetime)Author is leading expert in partial differential equations 1. Distribution Theory And Fourier Analysis. Lars Hörmander. Includes Bibliographical References And Index.