Fourier integral operators of infinite order on Gevrey spaces. Applications to the Cauchy problem for certain hyperbolic operators,

The authors study Fourier integral operators (FIO) with amplitude of infinite order (i.e., amplitude having a suitable exponential growth in the dual space of the variables). Theorem 2.10 deals with the composition of a pseudodifferential operator and an FIO of infinite order on Gevrey spaces. The action of FIOs of infinite order on the Gevrey wave front set of an ultradistribution is also investigated. In Section 3 a parametrix to the Cauchy problem for certain hyperbolic operators as an FIO of infinite order is constructed. In this way, results on semiglobal existence, uniqueness and propagation of Gevrey singularities of the solution to the Cauchy problem mentioned above are proved. The authors generalize some theorems of Taniguchi, Mizohata, and Boutet de Monvel.