Estimates for Fourier transforms of surface measures in ${\mathbb{R}}^3$ and PDE applications

Abstract

An explicit 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 ${\mathbb{R}}^3$, 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 $q$.