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

Author: Michael Greenblatt
Journal: Trans. Amer. Math. Soc. 368 (2016), 6601-6625
MSC (2010): Primary 42B20
Published electronically: November 12, 2015
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Abstract: We prove uniform estimates for the decay rate of the Fourier transform of measures supported on real-analytic hypersurfaces in $ {\bf {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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Additional Information

Michael Greenblatt
Affiliation: Department of Mathematics, Statistics, and Computer Science, University of Illinois at Chicago, 322 Science and Engineering Office, 851 S. Morgan Street, Chicago, Illinois 60607-7045

Received by editor(s): January 30, 2014
Received by editor(s) in revised form: May 14, 2014, and August 31, 2014
Published electronically: November 12, 2015
Additional Notes: This research was supported in part by NSF grant DMS-1001070
Article copyright: © Copyright 2015 American Mathematical Society