The Technique of Pseudodifferential Operators

TABLE OF CONTENTS......Page 8
0.0. Some special notations, used in the book......Page 14
0.1. The Fourier transform; elementary facts......Page 16
0.2. Fourier analysis for temperate distributions on In......Page 22
0.3. The Paley-Wiener theorem; Fourier transform for general uE D'......Page 27
0.4. The Fourier-Laplace method; examples......Page 33
0.5. Abstract solutions and hypo-ellipticity......Page 43
0.6. Exponentiating a first order linear differential operator......Page 44
0.7. Solving a nonlinear first order partial differential equation......Page 49
0.8. Characteristics and bicharacteristics of a linear PDE......Page 53
0.9. Lie groups and Lie algebras for classical analysts......Page 58
1.1. Definition of pdo's......Page 65
1.2. Elementary properties of ipdo's......Page 69
1.3. Hoermander symbols; Weyl pdo's; distribution kernels......Page 73
1.4. The composition formulas of Beals......Page 77
1.5. The Leibniz' formulas with integral remainder......Page 82
1.6. Calculus of 1pdo's for symbols of Hoermander type......Page 85
1.7. Strictly classical symbols; some lemmata for application......Page 91
2.0. Introduction......Page 94
2.1. Elliptic and md-elliptic Vdo's......Page 95
2.2. Formally hypo-elliptic pdo's......Page 97
2.3. Local md-ellipticity and local md-hypo-ellipticity......Page 100
2.4. Formally hypo-elliptic differential expressions......Page 104
2.5. The wave front set and its invariance under yxlo's......Page 106
2.6. Systems of ,do's......Page 110
3.1. L2-boundedness of zero-order do's......Page 112
3.2. L2-boundedness for the case of 6>0......Page 116
3.3. Weighted Sobolev spaces; K-parametrix and Green inverse......Page 119
3.4. Existence of a Green inverse......Page 126
3.5. Hs-compactness for ftpdo's of negative order......Page 130
4.0. Introduction......Page 131
4.1. Distributions and temperate distributions on manifolds......Page 132
4.2. Distributions on S-manifolds; manifolds with conical ends......Page 136
4.3. Coordinate invariance of pseudodifferential operators......Page 142
4.4. Pseudodifferential operators on S-manifolds......Page 147
4.5. Order classes and Green inverses on S-manifolds......Page 152
5.0. Introduction......Page 157
5.1. Elliptic problems in free space; a summary......Page 160
5.2. The elliptic boundary problem......Page 162
5.3. Conversion to an &n-problem of Riemann-Hilbert type......Page 167
5.4. Boundary hypo-ellipticity; asymptotic expansion mod av......Page 170
5.5. A system of fide's for the Vj of problem 3.4......Page 175
5.6. Lopatinskij-Shapiro conditions; normal solvability of (2.2).......Page 182
5.7. Hypo-ellipticity, and the classical parabolic problem......Page 187
5.8. Spectral and semi-group theory for ado's......Page 192
5.9. Self-adjointness for boundary problems......Page 199
5.10. C*-algebras of tpdo's; comparison algebras......Page 202
6.1. First order symmetric hyperbolic systems of PDE......Page 209
6.2. First order symmetric hyperbolic systems of fide's on n .......Page 213
6.3. The evolution operator and its properties......Page 219
6.4. N-th order strictly hyperbolic systems and symmetrizers.......Page 223
6.5. The particle flow of a single hyperbolic pde......Page 228
6.6. The action of the particle flow on symbols......Page 232
6.7. Propagation of maximal ideals and propagation of singularities......Page 236
7.0. Introduction......Page 239
7.1. Algebra of hyperbolic polynomials......Page 240
7.2. Hyperbolic polynomials and characteristic surfaces......Page 243
7.3. The hyperbolic Cauchy problem for variable coefficients......Page 248
7.4. The cone h for a strictly hyperbolic expression of type e?......Page 251
7.5. Regions of dependence and influence; finite propagation speed......Page 254
7.6. The local Cauchy problem; hyperbolic problems on manifolds......Page 257
8.0. Introduction......Page 260
8.1. ,do's as smooth operators of L(H0)......Page 261
8.2. The 11DO-theorem......Page 264
8.3. The other half of the gbbO-theorem......Page 270
8.4. Smooth operators; the V -algebra property; 'Wdo-calculus......Page 274
8.5. The operator classes 'S and 'IVL , and their symbols......Page 278
8.6 The Frechet algebras y"x0, and the Weinstein- Zelditch class......Page 284
8.7 Polynomials in x and ax with coefficients in 'TX......Page 288
8.8 Characterization of qtX by the Lie algebra......Page 292
9.0. Introduction......Page 295
9.1. Flow invariance of V10......Page 296
9.2. Invariance of Vsm under particle flows......Page 299
9.3. Conjugation of Optpx with eiKt , KE Opy)ce......Page 302
9.4. Coordinate and gauge invariance; extension to S-manifolds......Page 306
9.5. Conjugation with eiKt for a matrix-valued K=k(x,D)......Page 309
9.6. A technical discussion of commutator equations......Page 314
9.7. Completion of the proof of theorem 5.4......Page 318
10.0. Introduction......Page 323
10.1. A refinement of the concept of observable......Page 327
10.2. The invariant algebra and the Foldy-Wouthuysen transform......Page 332
10.3. The geometrical optics approach for the Dirac algebra P......Page 337
10.4. Some identities for the Dirac matrices......Page 342
10.5. The first correction z0 for standard observables......Page 347
10.6. Proof of the Foldy-Wouthuysen theorem......Page 356
10.7. Nonscalar symbols in diagonal coordinates of......Page 363
10.8. The full symmetrized first correction symbol zS......Page 369
10.9. Some final remarks......Page 380
References......Page 383
Index......Page 393