On the sharp constant in the Bianchi-Egnell stability inequality

TL;DR

This paper investigates the sharp constant in the Bianchi-Egnell stability inequality for fractional Sobolev spaces, showing it is strictly less than the spectral gap constant and cannot be approached by sequences converging to the optimizer manifold.

Contribution

It proves that the best constant in the fractional Bianchi-Egnell inequality is strictly smaller than the spectral gap constant, providing a precise expansion along test functions.

Findings

The sharp constant c_BE(s) is strictly less than the spectral gap constant.

Sequences approaching the optimizer manifold do not attain c_BE(s) asymptotically.

The proof uses a detailed expansion of the quotient along a converging sequence.

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 \[ c_{BE}(s) = \inf_{f \in \dot{H}^s(\mathbb R^d) \setminus \mathcal M} \frac{\|(-\Delta)^{s/2} f\|_{L^2(\mathbb R^d)}^2 - S_{d,s} \|f\|_{L^{2^*}(\mathbb R^d)}^2}{\text{dist}_{\dot{H}^s(\mathbb R^d)}(f, \mathcal M)^2} \] 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 $\mathcal M$.