Optimal regularity of Fourier integral operators with one-sided folds

We obtain optimal continuity in Sobolev spaces for the Fourier integral operators associated to singular canonical relations, when one of the two projections is a Whitney fold. The regularity depends on the type, $k$, of the other projection from the canonical relation ($k=1$ for a Whitney fold). We prove that one loses $(4+\frac{2}{k})^{-1}$ of a derivative in the regularity properties. The proof is based on the $L^2$ estimates for oscillatory integral operators.