INDEX 429 arding, 6, 73, 92–96, 101, 142, 145, 214–216 Gronwall, 117, 265 Hardy–Littlewood–Sobolev, 236 Minkowski, 239 Schur, 82, 314, 327 interpolation, 236, 241, 244 inverse, 91–92 approximate, 400 Inverse Function Theorem, 49, 400, 420 Jacobi’s identity, 20, 21 kernel Bergman, 293, 307–309 of Fredholm operator, 416 Schwartz, 59, 65, 81, 82, 175, 209, 210, 238, 263, 370, 399 Laplace–Beltrami operator, 157, 243, 351, 353, 357, 362, 378 Leibnitz rule, 150 Lidskii’s Theorem, 413 Lie derivative, 395 lifting, 18–19, 203 Liouville measure, 366 Littlewood–Paley theory, 158, 159, 161, 163, 167, 214 localization, 39, 153, 155, 188, 195 manifolds, 339–363 definition of, 339 PDE on, 353–362 pseudodifferential operators on, 345–352 Riemannian, 344–345 smooth functions on, 340 Maslov index, 264 matrices J, 15, 16, 64, 257 notation for, 384 symplectic, 252–253, 262 transition, 340 Mean Ergodic Theorem, 367 meromorphic family of operators, 110, 419 resolvents, 129, 131 microlocality, 195 microlocally invertible, 195 Morse Lemma, 46, 48–50 nondegeneracy condition, 15, 48, 155, 157, 237, 239, 240, 244, 282 nonnormal operators, 287 norm, 140, 346, 347, 371, 387, 411 normal forms, 273–289 complex symbols, 282–286 real symbols, 275–279 notation, 383–389 basic, 383–384 for estimates, 388 for functions, 385–387 for matrices, 384 for operators, 387 for sets, 384 multiindex, 385 observables, 3, 5, 8, 56, 247, 348, 370 Open Mapping Theorem, 417 order functions, 73 change of, 182, 183 definition, 72 examples, 72 log of, 182 order of vanishing, 148–152 oscillatory integral, 6, 40, 46, 52–54 oscillatory testing, 80 phase shift, 36 Planck’s constant, 1, 5 plurisubharmonic functions, 300–302 Poincar´ e’s Lemma, 230, 255, 396 on manifolds, 398 Poisson bracket, 4, 5, 20, 68, 106, 369, 386 polar decomposition, 252 principal symbol, 74, 213, 277, 279, 281, 282, 361, 371 principal type, 276, 278, 282 projection, 127–129, 131, 134, 368, 371, 373 Bergman, 293, 307 propagation of singularities, 279–281 pseudodifferential operators, 2, 4, 55–96 on manifold, 347 symbol of, 348, 349, 351 pseudolocality, 81, 204 pseudospectrum, 287, 288 push-forward, 246, 394 quadratic forms, 295 quantization and commutators, 61 composition, 66 Fourier decomposition, 65
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