In Hypo-Analytic Structures Francois Treves provides a systematic approach to the study of the differential structures on manifolds defined by systems of complex vector fields. Serving as his main examples are the elliptic complexes, among which the De Rham and Dolbeault are the best known, and the tangential Cauchy-Riemann operators. Basic geometric entities attached to those structures are isolated, such as maximally real submanifolds and orbits of the system. Treves discusses the existence, uniqueness, and approximation of local solutions to homogeneous and inhomogeneous equations and delimits their supports.
The contents of this book consist of many results accumulated in the last decade by the author and his collaborators, but also include classical results, such as the Newlander-Nirenberg theorem. The reader will find an elementary description of the FBI transform, as well as examples of its use. Treves extends the main approximation and uniqueness results to first-order nonlinear equations by means of the Hamiltonian lift.

Contenuti

	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

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