Table of Contents

Preface v

1 Introduction, Notation and Preliminaries 1

2 Differentiation of Integrals Depending on Parameters 5

3 The Convolution 9

4 The Fourier Transform 17

5 Tempered Distributions 27

6 Symbols, Pseudo-Differential Operators and Asymptotic Expansions 31

7 A Partition of Unity and Taylor's Formula 43

8 The Product of Two Pseudo-Differential Operators 55

9 The Formal Adjoint of a Pseudo-Differential Operator 61

10 The Parametrix of an Elliptic Pseudo-Differential Operator 69

11 L^p-Boundedness of Pseudo-Differential Operators 75

12 The Sobolev Spaces H^s,P, -∞ < s < ∞, 1 ≤ p < ∞ 87

13 Closed Linear Operators 95

14 Minimal and Maximal Pseudo-Differential Operators 99

15 Global Regularity of Elliptic Partial Differential Equations 105

16 Weak Solutions of Pseudo-Differential Equations 109

17 Garding's Inequality 113

18 Strong Solutions of Pseudo-Differential Equations 121

19 One-Parameter Semigroups Generated by Pseudo-Differential Operators 129

20 Fredholm Operators 143

21 Fredholm Pseudo-Differential Operators 155

22 Symmetrically Global Pseudo-Differential Operators 167

23 Spectral Invariant of Symmetrically Global Pseudo-Differential Operators 177

Bibliography 181

Index 183