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 $s\in (0,\frac{d}{2})$. We prove that in dimension $d\ge 2$ the best constant \[ c_{BE}(s) = \inf_{f \in \dot{H}^s(\mathbb R^d) \setminus \mathcal M} \frac{\|(-Δ)^{s/2} f\|_{L^2(\mathbb R^d)}^2 - S_{d,s} \|f\|_{L^{2^*}(\mathbb R^d)}^2}{\text{dist}_{\dot{H}^s(\mathbb R^d)}(f, \mathcal M)^2} \] 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 $\mathcal{M}$.

6 pages, comments welcome

References in corpus (1)

• Critical functions and blow-up asymptotics for the fractional Brezis--Nirenberg problem in low dimension

Cited by in corpus (3)

• Stability with explicit constants of the critical points of the fractional Sobolev inequality and applications to fast diffusion
• An exceptional property of the one-dimensional Bianchi-Egnell inequality
• Energy asymptotics and blow-up phenomena for biharmonic Brézis-Nirenberg problem