Decoupling for smooth surfaces in $\mathbb{R}^3$

TL;DR

This paper establishes decoupling inequalities for smooth surfaces in three-dimensional space, extending previous results to all polynomial graphs and confirming a conjecture by Bourgain, Demeter, and Kemp.

Contribution

It proves uniform decoupling inequalities for polynomial graphs and smooth surfaces in $\

Findings

Decoupling inequalities hold for all polynomial graphs of degree $d$ with uniform constants.

A decoupling inequality is established for every smooth surface in $\

It confirms a conjecture of Bourgain, Demeter, and Kemp regarding decoupling for smooth surfaces.

Abstract

For each $d\geq 0$, we prove decoupling inequalities in $\mathbb R^3$ for the graphs of all bivariate polynomials of degree at most $d$ with bounded coefficients, with the decoupling constant depending uniformly in $d$ but not the coefficients of each individual polynomial. As a consequence, we prove a decoupling inequality for (a compact piece of) every smooth surface in $\mathbb{R}^3$, which in particular solves a conjecture of Bourgain, Demeter and Kemp.