Van der Corput lemmas and Fourier transforms of irregular hypersurface measures
Michael Greenblatt · arXiv (Cornell University) · 2014
Using one-dimensional Van der Corput lemmas on functions that are either mixed homogeneous in several variables, or are arbitrary real-analytic functions in two variables, we prove estimates on Fourier transforms of hypersurface measures and also provide some PDE applications. The densities are allowed to have a range of singularities and in the mixed homogeneous case can be quite irregular. The local n-dimensional theorems have some global extensions, while the two-dimensional results for general real-analytic phase are all local and use a local resolution of singularities theorem in two dimensions which provides appropriate coordinate systems to effectively use one-dimensional Van der Corput lemmas.