Hypo-Analytic Structures

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