Sharp gradient stability for the Sobolev inequality

Abstract

We prove a sharp quantitative version of the $p$-Sobolev inequality for any $1<p<n$, with a control on the strongest possible distance from the class of optimal functions. Surprisingly, the sharp exponent is constant for $p<2$, while it depends on $p$ for $p>2$.

arXiv comment32 pages

Cited by in corpus (1)

• Stability in Gagliardo-Nirenberg-Sobolev inequalities: flows, regularity and the entropy method