Uniform Estimates for the Fourier Transform of Surface Carried Measures in R3 and an Application to Fourier Restriction

Let S be a hypersurface in R-3 which is the graph of a smooth, finite type function phi, and let mu = rho d sigma be a surface carried measure on S, where d sigma denotes the surface element on S and rho a smooth density with sufficiently small support. We derive uniform estimates for the Fourier transform (mu) over cap of mu, which are sharp except for the case where the principal face of the Newton polyhedron of phi, when expressed in adapted coordinates, is unbounded. As an application, we prove a sharp L-p-L-2 Fourier restriction theorem for S in the case where the original coordinates are adapted to phi. This improves on earlier joint work with M. Kempe.