On the sharp constant in the Bianchi-Egnell stability inequality
Tobias König
TL;DR
The paper proves that the sharp stability constant $c_{BE}(s)$ in the Bianchi–Egnell inequality for the fractional Sobolev inequality is strictly smaller than the spectral bound $\frac{4s}{d+2s+2}$ in dimensions $d \ge 2$ and exponents $s \in (0,\frac{d}{2})$. The authors achieve this by performing a refined asymptotic expansion of the Bianchi–Egnell quotient along a carefully chosen perturbation of the Talenti bubble, using the equality case of the associated spectral gap and a degree-2 spherical harmonic to ensure a positive cubic contribution. This establishes that the global best constant cannot be attained by sequences converging to the manifold of optimizers $\\mathcal{M}$, highlighting a nonlocal aspect of the stability phenomenon. The result extends the understanding of stability beyond the local analysis near $\\mathcal{M}$ and corroborates analogous phenomena in related geometric inequalities. The method combines explicit test-function expansions with spectral-gap information for fractional Sobolev spaces.
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 $d \geq 2$ 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 $\mathcal M$.