Lp → Lq regularity of fourier integral operators with caustics

The caustics of Fourier integral operators are defined as caustics of the corresponding Schwartz kernels (Lagrangian distributions on X x Y). The caustic set Sigma(C) of the canonical relation is characterized as the set of points where the rank of the projection pi : C --> X x Y is smaller than its maximal value, dim(X x Y)-1. We derive the L-p(Y) --> L-q(X) estimates on Fourier integral operators with caustics of corank 1 ( such as caustics of type A(m+1), m is an element ofN). For the values of p and q outside of a certain neighborhood of the line of duality, q = p', the L-p --> L-q estimates are proved to be caustics-insensitive. We apply our results to the analysis of the blow-up of the estimates on the half-wave operator just before the geodesic flow forms caustics.