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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