Fourier decay and $L^p$ Sobolev smoothing for weighted hypersurface measures in ${\mathbb R}^3$

Abstract

We consider local hypersurface measures in ${\mathbb R}^3$ whose density is allowed to have a weight function constructed from real analytic functions in a broad sense. We prove $L^p$ Sobolev smoothing theorems for convolutions with such surface measures and Fourier transform decay rate results for these measures, generalizing and subsuming earlier results for smooth densities. Our theorems are sharp in an appropriate sense and can be described in terms of relatively simple properties of the surfaces and weight functions.