Estimates for Fourier transforms of surface measures in R^3 and PDE applications
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.
Keywords
Cite
@article{arxiv.1302.4070,
title = {Estimates for Fourier transforms of surface measures in R^3 and PDE applications},
author = {Michael Greenblatt},
journal= {arXiv preprint arXiv:1302.4070},
year = {2014}
}
Comments
28 pages. v6: Explicitly described how one starts the resolution of singularities process in the beginning of section 3 and the beginning of the proof of Theorem 1.2; this was somewhat glossed over before