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Fourier integral operators. (English) Zbl 0841.35137
Progress in Mathematics (Boston, Mass.). 130. Basel: Birkhäuser. viii, 142 p. (1996).
As observed in the introduction, this volume is essentially a TEXed version of some lecture notes distributed by the Courant Institute of New York, corresponding to a course given by the author in the years 1970-71 about Fourier integral operators (FIO). Despite of many further results recently appeared, this book is still interesting, giving a quick and elegant introduction to the field, more adapted to non-specialists than, for example, the exposition in the monumental work of L. Hörmander [The analysis of linear partial differential operators, I–IV Springer-Verlag, Berlin (1983; Zbl 0521.35001, Zbl 0521.35002), (1985; Zbl 0601.35001, Zbl 0612.35001)].
The contents are the following: after preliminaries on distributions, Fourier transform and wave front set in Chapter 1, the local theory of FIO is discussed in Chapter 2, having as particular case the calculus of pseudo-differential operators. Chapter 3 is devoted to symplectic differential geometry: symplectic vector spaces, symplectic manifolds, Lagrangian manifolds and links with classical mechanics and variational calculus. Chapter 4 concerns the global theory of FIO; pseudo-differential operators on manifolds are discussed as particular cases. Finally, Chapter 5 presents two applications: the Cauchy problem for strictly hyperbolic equations and caustics in oscillatory integrals.
This last part of the book contains some interesting historical remarks, about connections with the work of Newton on optics and Huygens on wave fronts.
Reviewer: L.Rodino (Torino)

35S30 Fourier integral operators applied to PDEs
35-02 Research exposition (monographs, survey articles) pertaining to partial differential equations
37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems