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

Abstract

For each d ≥ 0, we prove decoupling inequalities in 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 R 3 , which in particular solves a conjecture of Bourgain, Demeter and Kemp.