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 s ∈ (0, d 2 ). We prove that in dimension d ≥ 2 the best constantis strictly smaller than the spectral gap constant 4s d+2s+2 associated to sequences which converge to the manifold 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 M.