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

TL;DR

This paper introduces new decay estimates for Fourier transforms of surface measures in R^3 using resolution of singularities and Van der Corput lemma, leading to improved PDE solution bounds without relying on adapted coordinates.

Contribution

It provides novel decay estimates for Fourier transforms of surface measures in R^3 and new proofs of existing estimates, utilizing resolution of singularities instead of adapted coordinate systems.

Findings

New decay rate estimates for Fourier transforms of surface measures.

L^q bounds for PDE solutions based on initial data norms.

Elimination of the need for adapted coordinate systems in analysis.

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.