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arXiv · 1302.4070

Estimates for Fourier transforms of surface measures in R^3 and PDE applications

Michael Greenblatt
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Abstract

A local two-dimensional resolution of singularities theorem and arguments based on the Van der Corput lemma are used to give new estimates for the decay rate of the Fourier transform of a locally defined smooth hypersurface measure in R^3, as well as to provide new proofs of some known estimates. These are then used to give L^q bounds on solutions to certain PDE problems in terms of the L^p norms of their initial data for various values of p and q. Unlike much of the earlier work in this subject, no use is made of the adapted coordinate systems that have been often been used to study two-dimensional oscillatory integrals; all of the needed information is furnished by the resolution of singularities theorem.

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Michael Greenblatt. 2013-02-17. Estimates for Fourier transforms of surface measures in R^3 and PDE applications. https://arxiv.org/abs/1302.4070

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