Sharp Quantitative Stability for the Affine \(p\)-Sobolev Inequality, Part I: The Case \(2\le p<n\)
arXiv:2606.09555
summary
The authors establish a sharp quantitative stability theorem for the affine L^p‑Sobolev inequality in the range 2 ≤ p < n, proving that the stability exponent is exactly p and cannot be improved.
Abstract
We prove a sharp quantitative stability result for the affine \(L^p\)-Sobolev inequality, for \(p\ge2\), introduced by Lutwak--Yang--Zhang (\emph{J. Differential Geom.}, \textbf{62} (2002), 17--38). Moreover, the stability exponent is shown to be optimal, and equal to \(p\).
Topics & keywords
#affine Sobolev inequality#quantitative stability#sharp constants#optimal exponent#functional inequalitiesaffine L^p-Sobolev inequalitystability exponentsharp quantitative stabilityLutwak-Yang-Zhang inequalityp≥2