$L\sp p$-$L\sp q$ 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 $\Sigma (C)$ of the canonical relation $C$ is characterized as the set of points where the rank of the projection $\pi :C\to X\times Y$ is smaller than its maximal value, $dim(X\times Y)-1$. We derive the $L\sp p(Y)\to L\sp q(X)$ estimates on Fourier integral operators with caustics of corank 1 (such as caustics of type $A\sb{m+1}$, $m\in \mathbb{N}$). For the values of $p$ and $q$ outside of certain neighborhood of the line of duality, $q=p^{\prime }$, the $L\sp p\to L\sp q$ 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.

24 pages, 1 figure