Sharp Stability for the Affine Fractional Sobolev Inequality

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.} $\mathbf{\text{388}}$ (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.