Introduction Chapter 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 Chapter Ⅱ. Definition and Basic Properties of Distributions Summary 2.1. Basic Definitions 2.2. Localization 2.3. Distributions with Compact Support Notes Chapter 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 Chapter 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 Chapter Ⅴ. Distributions in Product Spaces Summary 5.1. Tensor Products 5.2. The Kernel Theorem Notes Chapter 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 Chapter Ⅶ. The Fourier Transformation Summary 7.1. The Fourier Transformation in y and in y' 7.2. Poisson's Summation Formula and Periodic Distributions 7.3. The Fourier-Laplace Transformation in ε' 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 Holder Estimates Notes Chapter 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 Chapter 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 Bibliography Index Index of Notation
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