A note on strong-form stability for the Sobolev inequality

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 $| abla u - abla v|_{p}$, where $v$ is an extremal function for the Sobolev inequality.