On the sharp constant in the Bianchi–Egnell stability inequality

Abstract This note is concerned with the Bianchi–Egnell inequality, which quantifies the stability of the Sobolev inequality, and its generalization to fractional exponents . We prove that in dimension the best constant is strictly smaller than the spectral gap constant associated to sequences that converge to the manifold of Sobolev optimizers. In particular, 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 .