Uniform estimates for the Fourier transform of surface carried measures in $\Bbb R^3$ and an application to Fourier restriction

Abstract

Let $S$ be a hypersurface in $\Bbb R^3$ which is the graph of a smooth, finite type function $\phi,$ and let $\mu=\rho\, d\si$ be a surface carried measure on $S,$ where $d\si$ denotes the surface element on $S$ and $\rho$ a smooth density with suffiently small support. We derive uniform estimates for the Fourier transform $\hat \mu$ 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.