Asymptotics of the estimates on oscillatory integral operators with degenerate phase functions are considered. The main result is the estimate for the operator associated to the canonical relation which is a two-sided Whitney fold, asymptotically degenerate on one side.

The results are applied to the Radon Transform of Melrose and Taylor, the integral operator which arises in the classical scattering theory. We consider two particular cases: scattering on an obstacle admitting tangent planes with precise contact and on an obstacle with asymptotically small sectional curvature. In these cases the regularity properties of the Radon Transform of Melrose and Taylor easily follow from the asymptotic estimates for oscillatory integral operators.

We also derive the regularity properties of this Radon Transform for a general kind of convex compact obstacle with a smooth boundary. The regularity is formulated in terms of the highest order of contact of the light rays with the boundary of an obstacle.