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AuthorHรถrmander, Lars. author
TitleThe Analysis of Linear Partial Differential Operators I [electronic resource] : Distribution Theory and Fourier Analysis / by Lars Hรถrmander
ImprintBerlin, Heidelberg : Springer Berlin Heidelberg, 1998
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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 deยญ velopment has made it difficult to bring the book up to date. Howevยญ 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 differยญ ential equations up to that time, and it has provided an ideal frameยญ work for later developments. "Linear partial differential operators" beยญ gan with a brief summary of distribution theory for this was still unยญ familiar to many analysts 20 years ago. The presentation then proยญ ceeded directly to the most general results available on partial differยญ ential operators. Thus the reader was expected to have some prior faยญ miliarity with the classical theory although it was not appealed to exยญ plicitly. Today it may no longer be necessary to include basic distribuยญ tion theory but it does not seem reasonable to assume a classical background in the theory of partial differential equations since modยญ 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 -- 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 $\cal S$ and in $\cal S$โ{128}{153}, -- 7.2. Poissonโ{128}{153}s Summation Formula and Periodic Distributions -- 7.3. The Fourier-Laplace Transformation in ?โ{128}{153}, -- 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 Hรถlder 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 -- Index of Notation

Mathematics Topological groups Lie groups Mathematics Topological Groups Lie Groups


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