$l^2$ decoupling theorem for surfaces in $\mathbb{R}^3$

Abstract

We identify a new way to divide the $\delta$-neighborhood of surfaces $\mathcal{M}\subset\mathbb{R}^3$ into a finitely-overlapping collection of rectangular boxes $S$. We obtain a sharp $(l^2,L^p)$ decoupling estimate using this decomposition, for the sharp range of exponents $2\leq p\leq 4$. Our decoupling inequality leads to new exponential sum estimates where the frequencies lie on surfaces which do not contain a line.