Gradient stability for the Sobolev inequality: the case $p\ge 2$

We prove a strong form of the quantitative Sobolev inequality in ${\mathbb{R}}^n$ for $p\ge 2$, where the deficit of a function $u\in \dot W^{1,p} $controls$| \nabla u -\nabla v|_{L^p}$foranextremalfunction$v$ in the Sobolev inequality.