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This volume is a useful introduction to the subject of Fourier Integral Operators and is based on the author´s classic set of notes. Covering a range of topics from Hörmander´s exposition of the theory, Duistermaat approaches the subject from symplectic geometry and includes application to hyperbolic equations (= equations of wave type) and oscillatory asymptotic solutions which may have caustics. This text is suitable for mathematicians and (theoretical) physicists with an interest in (linear) partial differential equations, especially in wave propagation, rep. WKB-methods.

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Preface.- 0. Introduction.- 1. Preliminaries.- 1.1 Distribution densities on manifolds.- 1.2 The method of stationary phase.- 1.3 The wave front set of a distribution.- 2. Local Theory of Fourier Integrals.- 2.1 Symbols.- 2.2 Distributions defined by oscillatory integrals.- 2.3 Oscillatory integrals with nondegenerate phase functions.- 2.4 Fourier integral operators (local theory).- 2.5 Pseudodifferential operators in Rn.- 3. Symplectic Differential Geometry.- 3.1 Vector fields.- 3.2 Differential forms.- 3.3 The canonical 1- and 2-form T (X).- 3.4 Symplectic vector spaces.- 3.5 Symplectic differential geometry.- 3.6 Lagrangian manifolds.- 3.7 Conic Lagrangian manifolds.- 3.8 Classical mechanics and variational calculus.- 4. Global Theory of Fourier Integral Operators.- 4.1 Invariant definition of the principal symbol.- 4.2 Global theory of Fourier integral operators.- 4.3 Products with vanishing principal symbol.- 4.4 L2-continuity.- 5. Applications.- 5.1 The Cauchy problem for strictly hyperbolic differential operators with C-infinity coefficients.- 5.2 Oscillatory asymptotic solutions. Caustics.- References.

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From the reviews:

This book remains a superb introduction to the theory of Fourier integral operators. While there are further advances discussed in other sources, this book can still be recommended as perhaps the very best place to start in the study of this subject.
-SIAM Review

This book is still interesting, giving a quick and elegant introduction to the field, more adapted to nonspecialists.

-Zentralblatt MATH
The book is completed with applications to the Cauchy problem for strictly hyperbolic equations and caustics in oscillatory integrals. The reader should have some background knowledge in analysis (distributions and Fourier transformations) and differential geometry.
-Acta Sci. Math.

"Duistermaat´s Fourier Integral Operators had its genesis in a course the author taught at Nijmegen in 1970. ... For the properly prepared and properly disposed mathematical audience Fourier Integral Operators is a must. ... it is a very important book on a subject that is both deep and broad." (Michael Berg, The Mathematical Association of America, May, 2011)

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