Uniform bounds for Fourier transforms of surface measures in R^3 with nonsmooth density

Abstract

We prove uniform estimates for the decay rate of the Fourier transform of measures supported on real-analytic hypersurfaces in R^3. If the surface contains the origin and is oriented such that its normal at the origin is in the direction of the z-axis and if dS denotes the surface measure for this surface, then the measures under consideration are of the form K(x,y)g(z) dS where K(x,y)g(z) is supported near the origin and both K(x,y) and g(z) are allowed to have singularities. The estimates here generalize the previously known sharp uniform estimates for when K(x,y)g(z) is smooth. The methods used in this paper involve an explicit two-dimensional resolution of singularities theorem, iterated twice, coupled with Van der Corput-type lemmas.