An Introduction to Involutive Structures
An involutive sub-bundle V of the complexified tangent bundle TcM on a manifold M is called locally integrable if its annihilator in the complex cotangent bundle is locally generated by exact differentials. Many problems for locally integrable structures (including local and microlocal regularity and sets of unicity) can be solved using the approximation theorem proved by M. S. Baouendi and F. Treves. This book contains a systematic description of similar results, starting with basic notions of the theory of involutive and locally integrable structures (Chapter 1). The Baouendi-Treves approximation theorem for various function spaces is treated in Chapter 2. The unique continuation property and approximate solutions are discussed in Chapter 3. Properties of locally solvable vector fields are described in Chapter 4. A description and applications of the FBI transform are contained in Chapter 5. Boundary properties of solutions of locally integrable vector fields are studied in Chapter 6. A description of the differential complex associated to an involutive structure and its homological properties is given in the last two chapters. The book ends with an epilogue containing various recent results in the field. The whole book is carefully organized and the reader is expected to have a basic knowledge of real and complex analysis, distribution theory and its use in partial differential equations, and basic facts from several complex variables. The book will be very useful for students wanting to learn the subject and it also introduces interesting recent results for specialists.