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

Abstract

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 $\Vert \nabla u-\nabla v{\Vert }_{L^p}$ for an extremal function $v$ in the Sobolev inequality.

arXiv commentSome organizational changes made from previous version. Accepted for publication in J. Eur. Math. Soc. (JEMS)