Oscillatory integral operators with low-order degeneracies

Abstract

We prove sharp $L\sp 2$-estimates for oscillatory integral and Fourier integral operators for which the associated canonical relation $\mathscr {C}\subset T\sp \ast\Omega\sb L\times T\sp \ast\Omega\sb R$ projects to $T\sp \ast\Omega\sb L$ and to $T\sp \ast\Omega\sb R$ with corank 1 singularities of type $\leq 2$. This includes two-sided cusp singularities. Applications are given to operators with one-sided swallowtail singularities such as restricted X-ray transforms for well-curved line complexes in five dimensions.