Gradient stability for the Sobolev inequality: the case $pgeq 2$

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