A note on strong-form stability for the Sobolev inequality

Abstract

In this note, we establish a strong form of the quantitive Sobolev inequality in Euclidean space for $p\in (1,n)$. Given any function $u\in {\dot{W}}^{1,p}({\mathbb{R}}^n)$, the gap in the Sobolev inequality controls $\Vert \nabla u-\nabla v{\Vert }_p$, where $v$ is an extremal function for the Sobolev inequality.