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

TL;DR

This paper establishes a strong quantitative form of the Sobolev inequality for p≥2, showing that the deficit controls the difference in gradients between a function and an extremal function.

Contribution

It proves a new strong stability estimate for the Sobolev inequality in the case p≥2, linking the deficit to the gradient difference.

Findings

Quantitative Sobolev inequality with p≥2

Deficit controls gradient difference from extremal

Enhanced stability estimates for Sobolev inequality

Abstract

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