Introduction to Microlocal Analysis
Richard Melrose
Massachusetts Institute of Technology
E-mail address: rbm@math.mit.edu
0.5C; Revised: 18-9-2007; Run: December 15, 2007
1991 Mathematics Subject Classification. All over the shop
Contents
Preface
9
Introduction
11
Chapter 1. Tempered distributions and the Fourier transform
1.1. Schwartz test functions
1.2. Linear transformations
1.3. Tempered distributions
1.4. Two big theorems
1.5. Examples
1.6. Two little lemmas
1.7. Fourier transform
1.8. Differential operators
1.9. Radial compactification
1.10. Problems
13
13
15
16
17
18
19
21
24
25
26
Chapter 2. Pseudodifferential operators on Euclidean space
2.1. Symbols
2.2. Pseudodifferential operators
2.3. Composition
2.4. Reduction
2.5. Asymptotic summation
2.6. Residual terms
2.7. Proof of Composition Theorem
2.8. Quantization and symbols
2.9. Principal symbol
2.10. Ellipticity
2.11. Elliptic regularity and the Laplacian
2.12. L2 boundedness
2.13. Square root and boundedness
2.14. Sobolev boundedness
2.15. Polyhomogeneity
2.16. Topologies and continuity of the product
2.17. Linear invariance
2.18. Local coordinate invariance
2.19. Semiclassical limit
2.20. Smooth and holomorphic families
2.21. Problems
29
29
33
35
36
37
39
40
41
42
44
46
47
48
50
53
55
57
58
59
63
64
Chapter 3.
69
Residual, or Schwartz, algebra
3
4
CONTENTS
3.1.
3.2.
3.3.
3.4.
3.5.
3.6.
3.7.
The residual algebra
The augmented residual algebra
Exponential and logarithm
The residual group
Traces on the residual algebra
Fredholm determinant
Fredholm alternative
69
70
73
73
74
76
79
Chapter 4. Isotropic calculus
4.1. Isotropic operators
4.2. Fredholm property
4.3. The harmonic oscillator
4.4. L2 boundedness and compactness
4.5. Sobolev spaces
4.6. Representations
4.7. Symplectic invariance of the isotropic product
4.8. Metaplectic group
4.9. Complex order
4.10. Resolvent and spectrum
4.11. Residue trace
4.12. Exterior derivation
4.13. Regularized trace
4.14. Projections
4.15. Complex powers
4.16. Index and invertibility
4.17. Variation 1-form
4.18. Determinant bundle
4.19. Index bundle
4.20. Index formulæ
4.21. Isotropic essential support
4.22. Isotropic wavefront set
4.23. Isotropic FBI transform
4.24. Problems
81
81
84
86
89
90
92
93
95
102
102
103
106
107
108
108
108
110
112
113
113
113
113
113
113
Chapter 5. Microlocalization
5.1. Calculus of supports
5.2. Singular supports
5.3. Pseudolocality
5.4. Coordinate invariance
5.5. Problems
5.6. Characteristic variety
5.7. Wavefront set
5.8. Essential support
5.9. Microlocal parametrices
5.10. Microlocality
5.11. Explicit formulations
5.12. Wavefront set of KA
5.13. Hypersurfaces and Hamilton vector fields
5.14. Relative wavefront set
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117
118
118
119
120
121
122
123
124
125
126
127
127
129
CONTENTS
5.15.
5.16.
5.17.
5.18.
5.19.
5.20.
5.21.
5.22.
5.23.
5.24.
5.25.
5.26.
5.27.
5.28.
5.29.
Proof of Proposition 5.9
Hörmander’s propagation theorem
Elementary calculus of wavefront sets
Pairing
Multiplication of distributions
Projection
Restriction
Exterior product
Diffeomorphisms
Products
Pull-back
The operation F∗
Wavefront relation
Applications
Problems
5
133
136
137
138
140
140
142
143
144
146
147
149
151
152
153
Chapter 6. Pseudodifferential operators on manifolds
6.1. C ∞ structures
6.2. Form bundles
6.3. Pseudodifferential operators
6.4. The symbol calculus
6.5. Pseudodifferential operators on vector bundles
6.6. Hodge theorem
6.7. Sobolev spaces and boundedness
6.8. Pseudodifferential projections
6.9. Semiclassical algebra
6.10. Heat kernel
6.11. Resolvent
6.12. Complex powers
6.13. Problems
155
155
156
157
162
164
165
169
171
173
175
175
175
175
Chapter 7. Scattering calculus
7.1. Scattering pseudodifferential operators
177
177
Chapter 8. Elliptic boundary problems
Summary
Introduction
Status as of 4 August, 1998
8.1. Manifolds with boundary
8.2. Smooth functions
8.3. Distributions
8.4. Boundary Terms
8.5. Sobolev spaces
8.6. Dividing hypersurfaces
8.7. Rational symbols
8.8. Proofs of Proposition 8.7 and Theorem 8.1
8.9. Inverses
8.10. Smoothing operators
8.11. Left and right parametrices
179
179
179
179
179
180
183
184
187
188
190
191
191
192
194
6
CONTENTS
8.12.
8.13.
8.14.
8.15.
8.16.
8.17.
8.18.
8.19.
8.20.
8.21.
8.22.
Right inverse
Boundary map
Calderòn projector
Poisson operator
Unique continuation
Boundary regularity
Pseudodifferential boundary conditions
Gluing
Local boundary conditions
Absolute and relative Hodge cohomology
Transmission condition
195
196
197
198
198
198
198
200
200
200
200
Chapter 9. The wave kernel
9.1. Hamilton-Jacobi theory
9.2. Riemann metrics and quantization
9.3. Transport equation
9.4. Problems
9.5. The wave equation
9.6. Forward fundamental solution
9.7. Operations on conormal distributions
9.8. Weyl asymptotics
9.9. Problems
201
213
217
218
222
222
227
230
232
236
Chapter 10. K-theory
10.1. What do I need for the index theorem?
10.2. Odd K-theory
10.3. Computations
10.4. Vector bundles
10.5. Isotropic index map
10.6. Bott periodicity
10.7. Semiclassical quantization
10.8. Symplectic bundles
10.9. Thom isomorphism
10.10. Chern-Weil theory and the Chern character
10.11. Todd class
10.12. Stabilization
10.13. Delooping sequence
10.14. Looping sequence
10.15. C ∗ algebras
10.16. K-theory of an algeba
10.17. The norm closure of Ψ0 (X)
10.18. The index map
10.19. Problems
237
237
237
241
242
244
247
249
249
249
249
253
253
253
253
253
253
253
253
253
Chapter
11.1.
11.2.
11.3.
11.4.
255
255
256
261
264
11. Hochschild homology
Formal Hochschild homology
Hochschild homology of polynomial algebras
Hochschild homology of C ∞ (X)
Commutative formal symbol algebra
CONTENTS
11.5.
11.6.
11.7.
11.8.
11.9.
11.10.
11.11.
11.12.
Hochschild chains
Semi-classical limit and spectral sequence
The E2 term
Degeneration and convergence
Explicit cohomology maps
Hochschild holomology of Ψ−∞ (X)
Hochschild holomology of ΨZ (X)
Morita equivalence
7
265
265
267
271
272
272
272
272
Chapter 12. The index theorem and formula
12.1. Outline
12.2. Fibrations
12.3. Smoothing families
12.4. Elliptic families
12.5. Spectral sections
12.6. Analytic index
12.7. Topological index
12.8. Proof of index theorem
12.9. Chern character of the index bundle
Problems
273
273
273
275
277
277
277
277
277
277
277
Appendix A. Bounded operators on Hilbert space
Index of Mathematicians
279
280
Appendix.
281
Bibliography
Preface
This is a somewhat revised version of the lecture notes from various courses
taught at MIT, now for Fall 2007.
There are many people to thank, including:
Benoit Charbonneau
Sine Rikke Jensen
Edith Mooers
Mark Joshi
Raul Tataru
Jonathan Kaplan
Jacob Bernstein
Vedran Sohinger
Peter Speh
Andras Vasy
Kaveh Fouladgar
Fang Wang
Nikola Kamburov
9
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 [5]. For a more exhaustive treatment see Volume I of Hörmander’s treatise [9].
Analysis on manifolds: Most of what we need can be picked up from Munkres’
book [10] or Spivak’s little book [13].
11