An Introduction to Involutive Structures (New Mathematical Monographs)

Half-title......Page 3
Series-title......Page 4
Title......Page 5
Copyright......Page 6
Contents......Page 7
Preface......Page 11
I.1 Complex vector fields......Page 15
I.2 The algebraic structure of [omitted](Ω)......Page 18
I.3 Formally integrable structures......Page 19
I.4 Differential forms......Page 21
I.5 The Frobenius theorem......Page 25
I.6 Analytic structures......Page 28
I.8 Some special structures......Page 29
I.9 Locally integrable structures......Page 33
I.10 Local generators......Page 35
I.11 Local generators in analytic structures......Page 39
I.12 Integrability of complex and elliptic structures......Page 40
I.13 Elliptic structures in the real plane......Page 42
I.14 Compatible submanifolds......Page 46
I.15 Locally integrable CR structures......Page 50
I.16 A CR structure that is not locally integrable......Page 52
I.17 The Levi form on a formally integrable structure......Page 56
Appendix: Proof of the Newlander–Nirenberg theorem......Page 61
Notes......Page 65
II.1 The approximation theorem......Page 66
II.2 Distribution solutions......Page 77
II.3 Convergence in standard functional spaces......Page 83
II.4 Applications......Page 97
Notes......Page 113
III.1 Sussmann\'s orbits......Page 115
III.2 Propagation of support and global unique continuation......Page 122
III.3 The strong uniqueness property for locally integrable solutions......Page 134
III.4 Proof of Theorem III.3.15......Page 140
III.5 Uniqueness for approximate solutions......Page 146
III.6 Real-analytic structures in the plane......Page 154
III.7 Further applications of Sussmann\'s orbits......Page 160
Notes......Page 161
IV Local solvability of vector fields......Page 163
IV.1 Planar vector fields......Page 164
IV.2 Solvability in C∞......Page 190
IV.3 Vector fields in several variables......Page 198
IV.4 Necessary conditions for local solvability......Page 213
Notes......Page 228
V.1 Certain submanifolds of hypoanalytic manifolds......Page 232
V.2 Microlocal analyticity and the FBI transform......Page 240
V.3 Microlocal smoothness......Page 250
V.4 Microlocal hypoanalyticity and the FBI transform......Page 253
V.5 Application of the FBI transform to the C∞ wave front set of solutions of nonlinear PDEs......Page 257
V.6 Applications to edge-of-the-wedge theory......Page 268
V.7 Application to the F. and M. Riesz theorem......Page 277
Notes......Page 284
VI.1 Existence of a boundary value......Page 285
VI.2 Pointwise convergence to the boundary value......Page 295
VI.3 One-sided local solvability in the plane......Page 300
VI.4 The Hp property for vector fields......Page 303
Notes......Page 321
VII.1 The exterior derivative......Page 322
VII.2 The local representation of the exterior derivative......Page 323
VII.4 The differential complex associated with a formally integrable structure......Page 325
VII.5 Localization......Page 326
VII.6 Germ solvability......Page 327
VII.7 V-cohomology and local solvability......Page 329
VII.8 The Approximate Poincaré Lemma......Page 330
VII.9 One-sided solvability......Page 333
VII.10 Localization near a point at the boundary......Page 335
VII.11 One-sided approximation......Page 336
VII.12 A Mayer–Vietoris argument......Page 337
VII.13 Local solvability versus local integrability......Page 342
Notes......Page 344
VIII Local solvability in locally integrable structures......Page 345
VIII.2 Local solvability in the analytic category......Page 346
VIII.3 Elliptic structures......Page 347
VIII.4 The Box operator associated with D......Page 351
VIII.5 The intersection number......Page 354
VIII.6 The intersection number under certain geometrical assumptions......Page 357
VIII.7 A necessary condition for one-sided solvability......Page 360
VIII.8 The sufficiency of condition (★)0......Page 362
VIII.9 Proof of Proposition VIII.8.2......Page 364
VIII.10 Solvability for corank one analytic structures......Page 368
Notes......Page 371
1 The similarity principle and applications......Page 375
1.1 Application to uniqueness in the Cauchy problem......Page 376
1.2 Application to infinitesimal bendings of surfaces......Page 377
2 Mizohata structures......Page 378
2.1 Mizohata structures in higher-dimensional manifolds......Page 381
3 Hypoanalytic structures......Page 384
4 The local model for a hypoanalytic manifold......Page 385
5 The sheaf of hyperfunction solutions on a hypoanalytic manifold......Page 386
A.1 Multipliers in h1......Page 388
A.2 Commutators......Page 390
A.3 Change of variables......Page 392
Bibliography......Page 395
Index......Page 404