On the sharp constant in the Bianchi-Egnell stability inequality

This note is concerned with the Bianchi-Egnell inequality, which quantifies the stability of the Sobolev inequality, and its generalization to fractional exponents $s\in (0,\frac{d}{2})$. We prove that in dimension $d\ge 2$ the best constant $$c_{BE}(s)=\underset{f\in {\dot{H}}^s({\mathbb{R}}^d)\setminus \mathcal{M}}{\inf }\frac{\mathrm{\parallel }(-\mathrm{\Delta }{)}^{s\mathrm{/}2}f{\mathrm{\parallel }}_{L^2({\mathbb{R}}^d)}^2-S_{d,s}\mathrm{\parallel }f{\mathrm{\parallel }}_{L^{2^{\ast }}({\mathbb{R}}^d)}^2}{{\text{dist}}_{{\dot{H}}^s({\mathbb{R}}^d)}(f,\mathcal{M}{)}^2}$$ is strictly smaller than the spectral gap constant $\frac{4s}{d+2s+2}$ associated to sequences which converge to the manifold $\mathcal{M}$ of Sobolev optimizers. In particular, $c_{BE}(s)$ cannot be asymptotically attained by such sequences. Our proof relies on a precise expansion of the Bianchi-Egnell quotient along a well-chosen sequence of test functions converging to $\mathcal{M}$.