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 $$\Vert \nabla u -\nabla v\Vert _{p}$$, where v is an extremal function for the Sobolev inequality.