𝐿^{𝑝}→𝐿^{𝑞} 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 × Y X\times Y ). The caustic set Σ ( C ) \Sigma (\mathbf {C}) of the canonical relation is characterized as the set of points where the rank of the projection π : C → X × Y \pi :\mathbf {C}\to X\times Y is smaller than its maximal value, dim ⁡ ( X × Y ) − 1 \dim (X\times Y)-1 . We derive the L p ( Y ) → L q ( X ) L^ p(Y)\to L^ q(X) estimates on Fourier integral operators with caustics of corank 1 1 (such as caustics of type A m + 1 A_{m+1} , m ∈ N m\in \mathbb {N} ). For the values of p p and q q outside of a certain neighborhood of the line of duality, q = p ′ q=p’ , the L p → L q L^ p\to 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.