Contents xiii
57.1. Construction of the parametrix 296
57.2. Determination of the metric on the boundary 300
§58. Spectral theory of elliptic operators 301
58.1. The nonselfadjoint case 301
58.2. Trace class operators 302
58.3. The selfadjoint case 305
58.4. The case of a compact manifold 309
§59. The index of elliptic operators in
Rn
311
59.1. Properties of Fredholm operators 311
59.2. Index of an elliptic ψdo 313
59.3. Fredholm elliptic ψdo’s in
Rn
316
59.4. Elements of K-theory 317
59.5. Proof of the index theorem 321
§60. Problems 324
Chapter VIII. Fourier Integral Operators 329
Introduction to Chapter VIII 329
§61. Boundedness of Fourier integral operators (FIO’s) 330
61.1. The definition of a FIO 330
61.2. The boundedness of FIO’s 331
61.3. Canonical transformations 333
§62. Operations with Fourier integral operators 334
62.1. The stationary phase lemma 334
62.2. Composition of a ψdo and a FIO 335
62.3. Elliptic FIO’s 337
62.4. Egorov’s theorem 338
§63. The wave front set of Fourier integral operators 340
§64. Parametrix for the hyperbolic Cauchy problem 342
64.1. Asymptotic expansion 342
64.2. Solution of the eikonal equation 344
64.3. Solution of the transport equation 346
64.4. Propagation of singularities 348
§65. Global Fourier integral operators 349
65.1. Lagrangian manifolds 349
65.2. FIO’s with nondegenerate phase functions 350
65.3. Local coordinates for a graph of a canonical transformation 353
65.4. Definition of a global FIO 358
65.5. Construction of a global FIO given a global canonical
transformation 360
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