| Preface | |
I | Formally and Locally Integrable Structures. Basic Definitions | 3 |
I.1 | Involutive systems of linear PDE defined by complex vector fields. Formally and locally integrable structures | 5 |
I.2 | The characteristic set. Partial classification of formally integrable structures | 11 |
I.3 | Strongly noncharacteristic, totally real, and maximally real submanifolds | 16 |
I.4 | Noncharacteristic and totally characteristic submanifolds | 23 |
I.5 | Local representations | 27 |
I.6 | The associated differential complex | 32 |
I.7 | Local representations in locally integrable structures | 39 |
I.8 | The Levi form in a formally integrable structure | 46 |
I.9 | The Levi form in a locally integrable structure | 49 |
I.10 | Characteristics in real and in analytic structures | 56 |
I.11 | Orbits and leaves. Involutive structures of finite type | 63 |
I.12 | A model case: Tube structures | 68 |
II | Local Approximation and Representation in Locally Integrable Structures | 73 |
II.1 | The coarse local embedding | 76 |
II.2 | The approximation formula | 81 |
II.3 | Consequences and generalizations | 86 |
II.4 | Analytic vectors | 94 |
II.5 | Local structure of distribution solutions and of L-closed currents | 100 |
II.6 | The approximate Poincare lemma | 104 |
II.7 | Approximation and local structure of solutions based on the fine local embedding | 108 |
II.8 | Unique continuation of solutions | 115 |
III | Hypo-Analytic Structures. Hypocomplex Manifolds | 120 |
III.1 | Hypo-analytic structures | 121 |
III.2 | Properties of hypo-analytic functions | 128 |
III.3 | Submanifolds compatible with the hypo-analytic structure | 130 |
III.4 | Unique continuation of solutions in a hypo-analytic manifold | 137 |
III.5 | Hypocomplex manifolds. Basic properties | 145 |
III.6 | Two-dimensional hypocomplex manifolds | 152 |
| Appendix to Section III.6: Some lemmas about first-order differential operators | 159 |
III.7 | A class of hypocomplex CR manifolds | 162 |
IV | Integrable Formal Structures. Normal Forms | 167 |
IV.1 | Integrable formal structures | 168 |
IV.2 | Hormander numbers, multiplicities, weights. Normal forms | 174 |
IV.3 | Lemmas about weights and vector fields | 178 |
IV.4 | Existence of basic vector fields of weight - 1 | 185 |
IV.5 | Existence of normal forms. Pluriharmonic free normal forms. Rigid structures | 191 |
IV.6 | Leading parts | 198 |
V | Involutive Structures with Boundary | 201 |
V.1 | Involutive structures with boundary | 202 |
V.2 | The associated differential complex. The boundary complex | 209 |
V.3 | Locally integrable structures with boundary. The Mayer-Vietoris sequence | 219 |
V.4 | Approximation of classical solutions in locally integrable structures with boundary | 226 |
V.5 | Distribution solutions in a manifold with totally characteristic boundary | 228 |
V.6 | Distribution solutions in a manifold with noncharacteristic boundary | 235 |
V.7 | Example: Domains in complex space | 246 |
VI | Local Integrability and Local Solvability in Elliptic Structures | 252 |
VI.1 | The Bochner-Martinelli formulas | 253 |
VI.2 | Homotopy formulas for [actual symbol not reproducible] in convex and bounded domains | 258 |
VI.3 | Estimating the sup norms of the homotopy operators | 264 |
VI.4 | Holder estimates for the homotopy operators in concentric balls | 269 |
VI.5 | The Newlander-Nirenberg theorem | 281 |
VI.6 | End of the proof of the Newlander-Nirenberg theorem | 287 |
VI.7 | Local integrability and local solvability of elliptic structures. Levi flat structures | 291 |
VI.8 | Partial local group structures | 297 |
VI.9 | Involutive structures with transverse group action. Rigid structures. Tube structures | 303 |
VII | Examples of Nonintegrability and of Nonsolvability | 312 |
VII.1 | Mizohata structures | 314 |
VII.2 | Nonsolvability and nonintegrability when the signature of the Levi form is |n - 2| | 319 |
VII.3 | Mizohata structures on two-dimensional manifolds | 324 |
VII.4 | Nonintegrability and nonsolvability when the cotangent structure bundle has rank one | 330 |
VII.5 | Nonintegrability and nonsolvability in Lewy structures. The three-dimensional case | 337 |
VII.6 | Nonintegrability in Lewy structures. The higher-dimensional case | 343 |
VII.7 | Example of a CR structure that is not locally integrable but is locally integrable on one side | 348 |
VIII | Necessary Conditions for the Vanishing of the Cohomology. Local Solvability of a Single Vector Field | 352 |
VIII.1 | Preliminary necessary conditions for exactness | 354 |
VIII.2 | Exactness of top-degree forms | 358 |
VIII.3 | A necessary condition for local exactness based on the Levi form | 364 |
VIII.4 | A result about structures whose characteristic set has rank at most equal to one | 367 |
VIII.5 | Proof of Theorem VIII.4.1 | 373 |
VIII.6 | Applications of Theorem VII |