$L^p\to L^q$ regularity of Fourier integral operators with caustics

Abstract:

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