Oscillatory integrals and Fourier transforms of surface carried measures

Abstract:

We suppose that $S$ is a smooth hypersurface in ${{\mathbf {R}}^{n + 1}}$ with Gaussian curvature $\kappa$ and surface measure $dS$, $w$ is a compactly supported cut-off function, and we let ${\mu _\alpha }$ be the surface measure with $d{\mu _\alpha } = w{\kappa ^\alpha } dS$. In this paper we consider the case where $S$ is the graph of a suitably convex function, homogeneous of degree $d$, and estimate the Fourier transform ${\hat \mu _\alpha }$. We also show that if $S$ is convex, with no tangent lines of infinite order, then ${\hat \mu _\alpha }(\xi )$ decays as $|\xi {|^{ - n / 2}}$ provided $\alpha \geqslant [(n + 3)/2]$. The techniques involved are the estimation of oscillatory integrals; we give applications involving maximal functions.