Sharp Stability for the Affine Fractional Sobolev Inequality

TL;DR

This paper establishes a precise stability result for the affine fractional Sobolev inequality, identifying the kernel, spectral gap, and stability constants in the fractional Sobolev space.

Contribution

It provides the first sharp quantitative stability analysis for the affine fractional Sobolev inequality, including kernel identification and spectral gap determination.

Findings

Identified the kernel of the affine Hessian.

Determined the sharp local spectral gap.

Proved the global stability constant is strictly smaller than the local spectral value.

Abstract

In this paper, we prove a sharp quantitative stability result for the affine fractional \(L^2\)-Sobolev inequality in \(\dot H^s(\mathbb R^n)\), \(0<s<1\), introduced by Haddad--Ludwig (\emph{Math. Ann.} \textbf{388} (2024), 1091--1115). In particular, we identify the kernel of the affine Hessian, determine the sharp local spectral gap, and show that the optimal global stability constant is strictly smaller than the corresponding local spectral value.