Hypo-Analytic Structures

In Hypo-Analytic Structures Franois 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.

Originally published in 1993.

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Preface
I.1Involutive systems of linear PDE defined by complex vector fields. Formally and locally integrable structures
I.2The characteristic set. Partial classification of formally integrable structures
I.3Strongly noncharacteristic, totally real, and maximally real submanifolds
I.4Noncharacteristic and totally characteristic submanifolds
I.5Local representations
I.6The associated differential complex
I.7Local representations in locally integrable structures
I.8The Levi form in a formally integrable structure
I.9The Levi form in a locally integrable structure
I.10Characteristics in real and in analytic structures
I.11Orbits and leaves. Involutive structures of finite type
I.12A model case: Tube structures IILocal Approximation and Representation in Locally Integrable Structures
II.1The coarse local embedding
II.2The approximation formula
II.3Consequences and generalizations
II.4Analytic vectors
II.5Local structure of distribution solutions and of L-closed currents
II.6The approximate Poincare lemma
II.7Approximation and local structure of solutions based on the fine local embedding
II.8Unique continuation of solutions IIIHypo-Analytic Structures. Hypocomplex Manifolds
III.1Hypo-analytic structures
III.2Properties of hypo-analytic functions
III.3Submanifolds compatible with the hypo-analytic structure
III.4Unique continuation of solutions in a hypo-analytic manifold
III.5Hypocomplex manifolds. Basic properties
III.6Two-dimensional hypocomplex manifolds Appendix to Section
III.6: Some lemmas about first-order differential operators
III.7A class of hypocomplex CR manifolds IVIntegrable Formal Structures. Normal Forms
IV.1Integrable formal structures
IV.2Hormander numbers, multiplicities, weights. Normal forms
IV.3Lemmas about weights and vector fields
IV.4Existence of basic vector fields of weight - 1
IV.5Existence of normal forms. Pluriharmonic free normal forms. Rigid structures
IV.6Leading parts VInvolutive Structures with Boundary
V.1Involutive structures with boundary
V.2The associated differential complex. The boundary complex
V.3Locally integrable structures with boundary. The Mayer-Vietoris sequence
V.4Approximation of classical solutions in locally integrable structures with boundary
V.5Distribution solutions in a manifold with totally characteristic boundary
V.6Distribution solutions in a manifold with noncharacteristic boundary
V.7Example: Domains in complex space VILocal Integrability and Local Solvability in Elliptic Structures
VI.1The Bochner-Martinelli formulas
VI.2Homotopy formulas for [actual symbol not reproducible] in convex and bounded domains
VI.3Estimating the sup norms of the homotopy operators
VI.4Holder estimates for the homotopy operators in concentric balls
VI.5The Newlander-Nirenberg theorem
VI.6End of the proof of the Newlander-Nirenberg theorem
VI.7Local integrability and local solvability of elliptic structures. Levi flat structures
VI.8Partial local group structures
VI.9Involutive structures with transverse group action. Rigid structures. Tube structures VIIExamples of Nonintegrability and of Nonsolvability
VII.1Mizohata structures
VII.2Nonsolvability and nonintegrability when the signature of the Levi form is |n - 2|
VII.3Mizohata structures on two-dimensional manifolds
VII.4Nonintegrability and nonsolvability when the cotangent structure bundle has rank one
VII.5Nonintegrability and nonsolvability in Lewy structures. The three-dimensional case
VII.6Nonintegrability in Lewy structures. The higher-dimensional case
VII.7Example of a CR structure that is not locally integrable but is locally integrable on one side VIIINecessary Conditions for the Vanishing of the Cohomology. Local Solvability of a Single Vector Field
VIII.1Preliminary necessary conditions for exactness
VIII.2Exactness of top-degree forms
VIII.3A necessary condition for local exactness based on the Levi form
VIII.4A result about structures whose characteristic set has rank at most equal to one
VIII.5Proof of Theorem
VIII.4.1
VIII.6Applications of Theorem VII