$L^p$ Regularity Estimates for a Class of Integral Operators with Fold Blowdown Singularities

We prove sharp $L^p$ regularity results for a class of generalized Radon transforms for families of curves in a three-dimensional manifold associated to a canonical relation with fold and blowdown singularities. The proof relies on decoupling inequalities by Wolff and Bourgain-Demeter for plate decompositions of thin neighborhoods of cones and $L^2$ estimates for related oscillatory integrals.Comment: 31 pages. Reorganized the proof of Proposition 4. Revised version incorporating referee's suggestions. To appear in The Journal of Geometric Analysi