On decoupling and restriction estimates

Abstract

In this short note, we prove that the restriction conjecture for the (hyperbolic) paraboloid in ${\mathbb{R}}^d$ implies the $l^p$-decoupling theorem for the (hyperbolic) paraboloid in ${\mathbb{R}}^{2d-1}$. In particular, this gives a simple proof of the $l^p$ decoupling theorem for the (hyperbolic) paraboloid in ${\mathbb{R}}^3$.

5 pages