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

TL;DR

This paper establishes sharp uniform estimates for the Fourier transform of surface measures on certain hypersurfaces in three-dimensional space and applies these results to improve Fourier restriction theorems.

Contribution

It provides new uniform Fourier transform estimates for surface measures on finite type hypersurfaces and enhances Fourier restriction results in adapted coordinates.

Findings

Derived sharp uniform Fourier transform estimates for surface measures.

Proved a sharp $L^p$-$L^2$ Fourier restriction theorem in adapted coordinates.

Improved previous results on Fourier restriction for hypersurfaces in $R^3$.

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.