# Hypo-Analytic Structures: Local Theory (PMS-40)

François Treves
Pages: 524
https://www.jstor.org/stable/j.ctt7zvvvk

1. Front Matter
(pp. i-vi)
(pp. vii-xii)
3. Preface
(pp. xiii-2)
4. I Formally and Locally Integrable Structures. Basic Definitions
(pp. 3-72)

A look at the contents of this first chapter should make perfectly clear what its title suggests: this is the chapter in which the fundamental concepts of the theory of involutive and locally integrable structures are laid out. Due to the high density of definitions some of the sections (e.g., I.3, I.4) are for reference rather than for reading. Here we highlight the main concepts introduced in chapter I, in the hope of helping readers select those in which they might want to delve further.

First, of course, the notion ofinvolutive(orformally integrable)structureon a${\ell ^\infty }$manifold...

5. II Local Approximation and Representation in Locally Integrable Structures
(pp. 73-119)

Definite advantages accrue from local integrability: this is the message of chapter II. They flow from two sources: the Approximation Formula (section II.2) and its generalization to differential forms (and currents), the Approximate Poincaré Lemma (section II.6); the local structure of distribution solutions, and its generalization to currents that are closed in the differential complex of the structure (section II.5). Locally any distribution is the finite sum of derivatives of continuous functions. The statement remains valid if we insert the adjective “solution,” meaning that both the distribution and the continuous functions are annihilated by all smooth sections L of the...

6. III Hypo-Analytic Structures. Hypocomplex Manifolds
(pp. 120-166)

LetMbe a generic submanifold of a complex manifold${\hat M}$; “generic” means that the restrictions toMof any set of local coordinates${z_1}{\text{,}}...{\text{,}}{z_m}(m = {\dim _\mathbb{C}}\hat M)$have$\mathbb{C}$-linearly independent differentials (which requires${\dim _\mathbb{R}}M\underline > m$). The restrictions of the functions${z_1}$, toMform a set of first integrals (in some open subset U ofM) in a locally integrable structure onM, specifically, the CR structure inherited byMfrom the complex structure of${\hat M}$. Another set of first integrals in U (for the same CR structure),${w_1}{\text{,}}...{\text{,}}{w_m}$, could also be the restrictions of complex coordinates in${\hat M}$; but in general need...

7. IV Integrable Formal Structures. Normal Forms
(pp. 167-200)

The quest for local invariants of an involutive structure goes back to the early days of the theory of several complex variables, when H. Poincaré and E. Cartan first looked at the boundary of a strongly pseudoconvex domain in${\mathbb{C}^2}$. The standpoint, in the present book, is less focused. In this fourth chapter we consider a general involutive structure and extract new invariants from the Taylor expansion (at a given point 0)of the coefficients of the vector fields L, (j= 1, . . . ,n) that span the tangent structure bundleVover a neighborhood of 0. In such...

8. V Involutive Structures with Boundary
(pp. 201-251)

In the present chapter the concept of an involutive structure on an open manifold is extended to a manifold$\overline M$with boundary. What the extended concept ought to be is evident: one deals with two open manifolds,M, the interior of$\overline M$and$\partial M$, its boundary; each carries an involutive structure; the two structures “fit” on$\partial M$. In more precise language, we are given a vector subbundleVof the complex tangent bundle$\mathbb{C}T\overline M$over the whole manifold$\overline M$(boundary included), which satisfies the Frobenius formal integrability condition and whose pullback to the boundary is a vector bundle. Since the...

9. VI Local Integrability and Local Solvability in Elliptic Structures
(pp. 252-311)

Chapters VI and VII complement each other. Both are devoted to the study of noteworthy classes of involutive structures from the viewpoint of local solvability and local integrability. Chapter VI discusses the basic classes of structures in which both properties hold; chapter VII presents a number of examples in which at least one of them does not.

Local solvability usually concerns equations Lu=fin which the data, i.e., the right-hand sidesf, are L-closed one-forms. The concept can be generalized as local exactness, at any level, in the differential complex associated with the involutive structure under consideration (section...

10. VII Examples of Nonintegrability and of Nonsolvability
(pp. 312-351)

As announced in the introduction to chapter VI, the contents of the present chapter consist of a number of examples of involutive structures that are not locally integrable or in whichlocal solvabilitydoes not hold. Local solvability means that the inhomogeneous equations

${L_J}u = j = 1{\text{, }}{\text{. }}{\text{. }}{\text{. ,}}n$, (1)

admit a local (distribution) solution whatever choice one makes of the righthand sides, thesmoothfunctions${f_1}{\text{, }}{\text{. }}{\text{. }}{\text{. ,}}{f_n}$, provided, of course, these satisfy the compatibility conditions

${L_J}{f_k} = {L_k}f,j,k = 1{\text{, }}{\text{. }}{\text{. }}{\text{. ,}}n$. (2)

(We are assuming that the vector fields${L_j}$commute.)

The simplest structure in which solvability does not hold is the structure on${\mathbb{R}^2}$(with coordinatesx,t) defined...

11. VIII Necessary Conditions for the Vanishing of the Cohomology. Local Solvability of a Single Vector Field
(pp. 352-414)

The examples of nonsolvability (as well as those of nonintegrability) in chapter VII were “constructed.” In the search for conditions necessary for solvability or, more generally, necessary for exactness in the differential complex associated with the given involutive structure (see section 1.6), another approach is by way ofa priori estimates. These are shown to follow from the hypothesis of exactness, by a Functional Analysis argument that is a routine adaptation of a classical lemma of Hörmander [1]. The claim here is that if exactness holds at the (p, q) level, then the absolute value of the integral of$f\^v$...

12. IX FBI Transform in a Hypo-Analytic Manifold
(pp. 415-450)

To its discoverer the Fourier transform was a tool to analyze the solutions of a particular linear partial differential equation. Since then its range of application has not ceased to expand, and now encompasses the most diverse fields of mathematical analysis, pure and applied. Today linear PDE theory (as well as certain types of nonlinear differential equations) is but one of its areas of application. In the context of differential equations it shows up under a variety of guises, such, for instance, as the Laplace transform and the Fourier-Borel transform. Circa 1968 Brós and Iagolnitzer introduced a version particularly well...

13. X Involutive Systems of Nonlinear First-Order Differential Equations
(pp. 451-484)

How much of the theory of locally integrable structures can be carried over to systems of first-ordernonlinearpartial differential equations? The last chapter of the present volume gives the beginning of an answer to this question. In essence our approach is “microlocal”: an involutive system of first-order nonlinear PDE is a${\ell ^\infty }$submanifold Σ, submitted to a number of conditions, of the complexified one-jet bundle$\mathbb{C}{g^1}M$over the manifoldM(the equations are set inM). The base projection must map Σ ontoM. Unless the PDE under consideration are linear, the intersection of Σ with a fibre of...

14. References
(pp. 485-492)
15. Index
(pp. 493-497)