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

Abstract In this paper, for d 1 d ≥ 1 and s (0, d2) s ∈ (0, d 2), we study the Bianchi–Egnell quotient aligned {Q} (f) = ₅ ₇̇⌁ ({ₑᵈ) {B}} (-) ^s/2 f ₋ℂ ({ {ₑᵈ) }² - S₃, ₒ f ₋^₂₃{₃-₂ₒ (Rᵈ) }²} {dist₇̇⌁ ({ₑᵈ) } (f, {B}) ²}, f Ḣˢ ({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 S₃, ₒ 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 Ḣˢ ({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.