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

In this paper, for d ≥ 1 d ≥ 1 and s ∈ (0,d/2) s ∈ ( 0 , d 2 ) , we study the Bianchi–Egnell quotient $${aligned} {Q}(f) = inf _{f ∈ Ḣ^s({R}^d) {B}} { (-Δ )s/2 f _{L^2({R}^d)}^2 - Sd,s f _{L2d/d-2s( R^d)}^2}{dist_{Ḣ^s({R}^d)}(f, {B})^2}, f ∈ Ḣ^s({R}^d) {B}, {aligned}$$ 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 $$Sd,s$$ S d , s is the best Sobolev constant and $${B}$$ 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 $${B}$$ B on which the quotient $${Q}(f)$$ Q ( f ) is larger than the lowest value attainable by sequences converging to $${B}$$ B . This behavior is surprising because it is contrary to the situation in dimension $$d ≥ 2$$ 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 , $${Q}(f)$$ Q ( f ) has no minimizer on $$Ḣ^s({R}^d) {B}$$ H ˙ s ( R d ) \ B , which again would be contrary to the situation in $$d ≥ 2$$ 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 $$d ≥ 1$$ d ≥ 1 . For $$d ≥ 2$$ 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.