Integral Operators with Singular Canonical Relations

We derive certain estimates and asymptotics for oscillatory integral operators with degenerate phase functions, concentrating our attention on the situation when one of the projections from the associated canonical relation is a Whitney fold. We discuss related topics: convexity, higher order degeneracy of canonical relations, and almost orthogonality. In the second part of the paper, we apply the developed technique to the Radon Transform of Melrose-Taylor, a particular degenerate Fourier integral operator which arises in the theory of scattering. We obtain its regularity properties for the scattering on a convex compact domain with a smooth boundary. The result is formulated in terms of the highest order of contact of lines with the boundary. 1 Oscillatory integral operators with singularities 1.