Author | Hรถrmander, Lars. author |
---|---|
Title | The Analysis of Linear Partial Differential Operators I [electronic resource] : Distribution Theory and Fourier Analysis / by Lars Hรถrmander |
Imprint | Berlin, Heidelberg : Springer Berlin Heidelberg, 1998 |
Connect to | http://dx.doi.org/10.1007/978-3-642-96750-4 |
Descript | online resource |
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, -- 7.2. Poissonโs Summation Formula and Periodic Distributions -- 7.3. The Fourier-Laplace Transformation in ?โ, -- 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