Uniform bounds for Fourier transforms of surface measures in R³ with nonsmooth density

Greenblatt, Michael

doi:10.1090/tran/6486

Uniform bounds for Fourier transforms of surface measures in R$^3$ with nonsmooth density
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Trans. Amer. Math. Soc. 368 (2016), 6601-6625 Request permission

Abstract:

We prove uniform estimates for the decay rate of the Fourier transform of measures supported on real-analytic hypersurfaces in $\textbf {{R}}^3$. 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.

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