An exceptional property of the one-dimensional Bianchi–Egnell inequality

In this paper, for \(d \ge 1\) and \(s \in (0,\frac{d}{2})\), we study the Bianchi–Egnell quotient

$$\begin{aligned} {\mathcal {Q}}(f) = \inf _{f \in \dot{H}^s({\mathbb {R}}^d) \setminus {\mathcal {B}}} \frac{\Vert (-\Delta )^{s/2} f\Vert _{L^2({\mathbb {R}}^d)}^2 - S_{d,s} \Vert f\Vert _{L^{\frac{2d}{d-2s}}(\mathbb R^d)}^2}{\text {dist}_{\dot{H}^s({\mathbb {R}}^d)}(f, {\mathcal {B}})^2}, \qquad f \in \dot{H}^s({\mathbb {R}}^d) \setminus {\mathcal {B}}, \end{aligned}$$

where \(S_{d,s}\) is the best Sobolev constant and \({\mathcal {B}}\) is the manifold of Sobolev optimizers. By a fine asymptotic analysis, we prove that when \(d = 1\), there is a neighborhood of \({\mathcal {B}}\) on which the quotient \({\mathcal {Q}}(f)\) is larger than the lowest value attainable by sequences converging to \({\mathcal {B}}\). This behavior is surprising because it is contrary to the situation in dimension \(d \ge 2\) described recently in König (Bull Lond Math Soc 55(4):2070–2075, 2023). This leads us to conjecture that for \(d = 1\), \({\mathcal {Q}}(f)\) has no minimizer on \(\dot{H}^s({\mathbb {R}}^d) \setminus {\mathcal {B}}\), which again would be contrary to the situation in \(d \ge 2\). As a complement of the above, we study a family of test functions which interpolates between one and two Talenti bubbles, for every \(d \ge 1\). For \(d \ge 2\), this family yields an alternative proof of the main result of König (Bull Lond Math Soc 55(4):2070–2075, 2023). For \(d =1\) we make some numerical observations which support the conjecture stated above.