Introduction to Microlocal Analysis

Richard Melrose

Author address: Massachusetts Institute of Technology

E-mail address: rbm@math.mit.edu

1991 Mathematics Subject Classification. All over the shop.

Contents

Preface 7

Introduction 9

Chapter 1. Tempered distributions and Fourier transform 11 1.1. Schwartz test functions 11 1.2. Tempered distributions 12 1.3. Two big theorems 14 1.4. Examples 14 1.5. Two little lemmas 15 1.6. Fourier transform 17 1.7. Differential operators 19 1.8. Radial compactification 21 1.9. Problems 21

Chapter 2. Pseudodifferential operators 25 2.1. Symbols 25 2.2. Pseudodifferential operators 28 2.3. Composition 30 2.4. Reduction 30 2.5. Asymptotic summation 31 2.6. Residual terms 33 2.7. Proof of Theorem 2.2 35 2.8. Problems 36

Chapter 3. Ellipticity and boundedness 37 3.1. Quantization and symbols 37 3.2. Ellipticity 39 3.3. The Laplacian 41 3.4. L^{2} boundedness 42 3.5. Residual terms 42 3.6. Square root 43 3.7. Proof of Proposition 3.1 44 3.8. Sobolev boundedness 45 3.9. Consequences 47 3.10. Calculus of supports 48 3.11. Singular supports 49 3.12. Pseudolocality 49 3.13. Coordinate invariance 50 3.14. Problems 52 Chapter 4. Wavefront set 55

4.1. Characteristic variety 55 4.2. Wavefront set 55 4.3. Essential support 57 4.4. Microlocal parametrices 58 4.5. Microlocality 59 4.6. Explicit formulations 60 4.7. Wavefront set of K_{A} 60 4.8. Elementary calculus of wavefront sets 61 4.9. Pairing 61 4.10. Multiplication of distributions 63 4.11. Projection 63 4.12. Restriction 65 4.13. Exterior product 67 4.14. Diffeomorphisms 67 4.15. Products 69 4.16. Pull-back 70 4.17. The operation F_{∗} 72 4.18. Wavefront relation 74 4.19. Applications 76 4.20. Problems 76

Chapter 5. Manifolds 79 5.1. C^{∞} structures 79 5.2. Form bundles 80 5.3. Pseudodifferential operators 81 5.4. Pseudodifferential operators on vector bundles 85 5.5. Laplacian on forms 87 5.6. Hodge theorem 87 5.7. Heat kernel 87 5.8. Resolvent 87 5.9. Complex powers 87

Chapter 6. The wave kernel 89 6.1. Hamilton-Jacobi theory 101 6.2. Problems 105 6.3. Riemann metrics and quantization 106 6.4. Transport equation 106 6.5. Problems 110 6.6. The wave equation 110 6.7. Forward fundamental solution 115 6.8. Operations on conormal distributions 118 6.9. Weyl asymptotics 120

Chapter 7. K-theory 125 7.1. Vector bundles 125 7.2. The ring K(X) 125 7.3. Chern-Weil theory and the Chern character 125 7.4. K_{1}(X) and the odd Chern character 125 7.5. C^{∗} algebras 125 7.6. K-theory of an algeba 125 7.7. The norm closure of Ψ^{0}(X) 125 7.8. The index map 125 7.9. Problems 125

Chapter 8. Hochschild homology 127 8.1. Formal Hochschild homology 127 8.2. Hochschild homology of polynomial algebras 128 8.3. Hochschild homology of C^{∞}(X) 133 8.4. Commutative formal symbol algebra 137 8.5. Hochschild chains for Ψ_{phg}^{ℤ}(X)/Ψ_{phg}^{−∞}(X) 137 8.6. Semi-classical limit and spectral sequence 138 8.7. The E_{2} term 139 8.8. Degeneration and convergence 144 8.9. Explicit cohomology maps 144 8.10. Hochschild holomology of Ψ^{−∞}(X) 144 8.11. Hochschild holomology of Ψ^{ℤ}(X) 144 8.12. Morita equivalence 144

Chapter 9. The index formula 145

Bibliography 147

6 CONTENTS

Preface

This is a somewhat revised version of the lecture notes from a course in 1990; in particular I have included the corrections made by Mark Joshi (thanks Mark, hope you are having fun in Dear Old Blighty.) I am continuing to correct and update these notes, thanks to Edie Mooers, ....

Here is the earlier preface.

I’m sorry to have to admit that I did not get a chance to even properly proofread large parts of these notes, especially Lecture XVI onward, and Lecture XV is essentially unwritten. They are offered for what they are worth as a ‘conventional’ course on pseudodifferential operators. Comments gratefully received, but there is absolutely no chance of my doing anything much with these until at least February 1991, if then.

8 PREFACE

Introduction

I shall assume some familiarity with distribution theory, with basic analysis and functional analysis and with (local) differential geometry. A passing knowledge of the theory of manifolds would also be useful. Any one or two of these prerequisites can be easily picked up along the way, but the prospective student with none of them should perhaps do some preliminary reading:

Distributions: I good introduction is Friedlander’s book [2]. For a more exhaustive treatment see Volume I of Hörmander’s treatise [3].

Analysis on manifolds: Most of what we need can be picked up from Munkres’ book [4] or Spivak’s little book [5].

10 INTRODUCTION 146 9. THE INDEX FORMULA

Bibliography

[1] J.-L. Brylinski and E. Getzler, The homology of algebras of pseudo-differential symbols and the noncommutative residue, K-theory 1 (1987), 385–403.

[2] F.G. Friedlander, Introduction to the theory of distributions, Cambridge University Press, Cambridge, London, New York, Melbourne, ?

[3] L. Hörmander, The analysis of linear partial differential operators, vol. 1, Springer-Verlag, Berlin, Heidelberg, New York, Tokyo, 1983.

[4] J.R. Munkres, Analysis on manifolds, ??, ??

[5] M Spivak, Calculus on manifolds, ??, ??