Hypo-analytic Structures (pms-40) (ebook)

Argumento de Hypo-analytic Structures (pms-40) (ebook)

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.

The Princeton Legacy Library uses the latest print-on-demand technology to again make available previously out-of-print books from the distinguished backlist of Princeton University Press. These paperback editions preserve the original texts of these important books while presenting them in durable paperback editions. The goal of the Princeton Legacy Library is to vastly increase access to the rich scholarly heritage found in the thousands of books published by Princeton University Press since its founding in 1905.0Preface 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