Fourier integral operators defined by classical symbols with exit behaviour

We present the detailed construction of the classical version of the calculus for Fourier Integral Operators in the class of symbols with exit behaviour (SG-symbols). In particular, we analyse what happens when one restricts the choice of amplitude and phase functions to the subset of the classical SG symbols. It turns out that the main composition theorem, obtained in the environment of general SG-classes, has a classical counterpart. As an application, we study the Cauchy problem for classical hyperbolic operators of order (1,1), refining the known results about the analogous problem for general SG-hyperbolic operators. The theory developed here will be used in forthcoming papers to study the propagation of singularities and the Weyl formula for suitable classes of operators defined on manifolds with cylindrical ends.