Theory of Distributions

Preface to the Second Edition
Preface to the First Edition
Contents
1 Introduction
	1.1 The Spaces C0∞ and S
	1.2 The Lp Spaces
		1.2.1 Definition
		1.2.2 The Inequalities of Hölder and Minkowski
		1.2.3 Some Properties
		1.2.4 The Riesz–Fischer Theorem
		1.2.5 Separability
		1.2.6 Duality
		1.2.7 General Lp Spaces
	1.3 The Convolution of Locally Integrable Functions
	1.4 Cones in Rn
	1.5 Advanced Practical Problems
	1.6 Notes and References
2 Generalities on Distributions
	2.1 Definitions
	2.2 Order of a Distribution
	2.3 Change of Variables
	2.4 Sequences and Series
	2.5 Support
	2.6 Singular Support
	2.7 Measures
	2.8 Multiplying Distributions by C∞ Functions
	2.9 Advanced Practical Problems
	2.10 Notes and References
3 Differentiation
	3.1 Derivatives
	3.2 The Local Structure of Distributions
	3.3 The Primitive of a Distribution
	3.4 Simple and Double Layers on Surfaces
	3.5 Advanced Practical Problems
	3.6 Notes and References
4 Homogeneous Distributions
	4.1 Definition
	4.2 Properties
	4.3 Advanced Practical Problems
	4.4 Notes and References
5 The Direct Product of Distributions
	5.1 Definition
	5.2 Properties
	5.3 Advanced Practical Problems
	5.4 Notes and References
6 Convolutions
	6.1 Definition
	6.2 Properties
	6.3 Existence
	6.4 The Convolution Algebras D'(Γ+) and D'(Γ)
	6.5 Regularization of Distributions
	6.6 Fractional Differentiation and Integration
	6.7 Advanced Practical Problems
	6.8 Notes and References
7 Tempered Distributions
	7.1 Definition
	7.2 Direct Product
	7.3 Convolution
	7.4 Advanced Practical Problems
	7.5 Notes and References
8 Integral Transforms
	8.1 The Fourier Transform in S(Rn)
	8.2 The Fourier Transform in S'(Rn)
	8.3 Properties of the Fourier Transform in S'(Rn)
	8.4 The Fourier Transform of Distributions with Compact Support
	8.5 The Fourier Transform of Convolutions
	8.6 The Laplace Transform
		8.6.1 Definition
		8.6.2 Properties
	8.7 Advanced Practical Problems
	8.8 Notes and References
9 Fundamental Solutions
	9.1 Definition and Properties
	9.2 Fundamental Solutions of Ordinary Differential Operators
	9.3 Fundamental Solution of the Heat Operator
	9.4 Fundamental Solution of the Laplace Operator
	9.5 Advanced Practical Problems
	9.6 Notes and References
10 Sobolev Spaces
	10.1 Definitions
	10.2 Elementary Properties
	10.3 Approximation by Smooth Functions
	10.4 Extensions
	10.5 Traces
	10.6 Sobolev Inequalities
	10.7 The Space H-s
	10.8 Advanced Practical Problems
	10.9 Notes and References
References
Index