Fourier transforms of irregular mixed homogeneous hypersurface measures

TL;DR

This paper establishes decay estimates for Fourier transforms of irregular mixed homogeneous hypersurface measures using Van der Corput lemmas, providing sharp local results that can be extended globally.

Contribution

It introduces new decay estimates for Fourier transforms of irregular hypersurface measures, leveraging Van der Corput lemmas, with results that are sharp within certain index ranges.

Findings

Decay estimates proven for Fourier transforms of irregular measures

Results are sharp for specific index ranges

Local decay estimates can be extended to global theorems

Abstract

With the help of Van der Corput lemmas, decay estimates are proven for Fourier transforms of mixed homogeneous hypersurface measures with densities that can be quite irregular. The primary results are local in nature, but can be extended to global theorems in an appropriate sense. The estimates are sharp for a certain range of indices in the theorems.