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

Abstract

AbstractIn this paper, for $$d \ge 1$$ d ≥ 1 and $$s \in (0,\frac{d}{2})$$ s ∈ ( 0 , d 2 ) , we study the Bianchi–Egnell quotient $$\begin{aligned} {\mathcal {Q}}(f) = \inf _{f \in \dot{H}^s({\mathbb {R}}^d) \setminus {\mathcal {B}}} \frac{\Vert (-\Delta )^{s/2} f\Vert _{L^2({\mathbb {R}}^d)}^2 - S_{d,s} \Vert f\Vert _{L^{\frac{2d}{d-2s}}(\mathbb R^d)}^2}{\text {dist}_{\dot{H}^s({\mathbb {R}}^d)}(f, {\mathcal {B}})^2}, \qquad f \in \dot{H}^s({\mathbb {R}}^d) \setminus {\mathcal {B}}, \end{aligned}$$ Q ( f ) = inf f ∈ H ˙ s ( R d ) \ B ‖ ( - Δ ) s / 2 f ‖ L 2 ( R d ) 2 - S d , s ‖ f ‖ L 2 d d - 2 s ( R d ) 2 dist H ˙ s ( R d ) ( f , B ) 2 , f ∈ H ˙ s ( R d ) \ B , where $$S_{d,s}$$ S d , s is the best Sobolev constant and $${\mathcal {B}}$$ B is the manifold of Sobolev optimizers. By a fine asymptotic analysis, we prove that when $$d = 1$$ d = 1 , there is a neighborhood of $${\mathcal {B}}$$ B on which the quotient $${\mathcal {Q}}(f)$$ Q ( f ) is larger than the lowest value attainable by sequences converging to $${\mathcal {B}}$$ B . This behavior is surprising because it is contrary to the situation in dimension $$d \ge 2$$ d ≥ 2 described recently in König (Bull Lond Math Soc 55(4):2070–2075, 2023). This leads us to conjecture that for $$d = 1$$ d = 1 , $${\mathcal {Q}}(f)$$ Q ( f ) has no minimizer on $$\dot{H}^s({\mathbb {R}}^d) \setminus {\mathcal {B}}$$ H ˙ s ( R d ) \ B , which again would be contrary to the situation in $$d \ge 2$$ d ≥ 2 . As a complement of the above, we study a family of test functions which interpolates between one and two Talenti bubbles, for every $$d \ge 1$$ d ≥ 1 . For $$d \ge 2$$ d ≥ 2 , this family yields an alternative proof of the main result of König (Bull Lond Math Soc 55(4):2070–2075, 2023). For $$d =1$$ d = 1 we make some numerical observations which support the conjecture stated above.