# A note on strong-form stability for the Sobolev inequality

## TL;DR

This paper proves a strong quantitative version of the Sobolev inequality in Euclidean space, linking the inequality gap to the gradient difference from extremal functions for functions in the Sobolev space.

## Contribution

It establishes a strong form of the quantitative Sobolev inequality, providing a precise control of the gradient difference in terms of the inequality gap.

## Key findings

- The gap in the Sobolev inequality controls the gradient difference from extremal functions.
- The result applies to functions in the homogeneous Sobolev space rac{1}{p}(rac{1}{n})
- The paper extends the understanding of stability in Sobolev inequalities.

## Abstract

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