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

Abstract

For each $d\ge 0$, 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.

Accepted by American Journal of Mathematics in June 2023; incorporated referee's comments, and updated references. This version is final