Hypo-Analytic Structures (PMS-40), Volume 40: Local Theory (PMS-40)
516Hypo-Analytic Structures (PMS-40), Volume 40: Local Theory (PMS-40)
516Hardcover
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Overview
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 editions preserve the original texts of these important books while presenting them in durable paperback and hardcover 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.
Product Details
ISBN-13: | 9780691635415 |
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Publisher: | Princeton University Press |
Publication date: | 04/19/2016 |
Series: | Princeton Mathematical Series , #50 |
Pages: | 516 |
Product dimensions: | 6.40(w) x 9.30(h) x 1.70(d) |
Table of Contents
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