On the sharp constant in the Bianchi–Egnell stability inequality

This note is concerned with the Bianchi–Egnell inequality, which quantifies stability of Sobolev and its generalization to fractional exponents s ∈ ( 0 , d 2 ) $s \in (0, \frac{d}{2})$ . We prove that in dimension ⩾ $d \geqslant 2$ best constant c B E = inf f H ̇ R ∖ M ∥ − Δ / L S ∗ dist $$\begin{equation*} c_{BE}(s) \inf _{f \dot{H}^s(\mathbb {R}^d) \setminus \mathcal {M}} \frac{\Vert (-\Delta )^{s/2} f\Vert _{L^2(\mathbb {R}^d)}^2 - S_{d,s} \Vert _{L^{2^*}(\mathbb {R}^d)}^2}{\operatorname{dist}_{\dot{H}^s(\mathbb {R}^d)}(f, {M})^2} \end{equation*}$$ strictly smaller than spectral gap 4 + $\frac{4s}{d+2s+2}$ associated sequences converge manifold $\mathcal {M}$ optimizers. In particular, $c_{BE}(s)$ cannot be asymptotically attained by such sequences. Our proof relies on a precise expansion quotient along well-chosen sequence test functions converging