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

TL;DR

This work analyzes the stability landscape of the Sobolev inequality in fractional order, focusing on the Bianchi-Egnell quotient and the role of dimension. By transporting the problem to the sphere via stereographic projection and exploiting the conformal symmetry, the authors derive a BE-local lower bound in 1D with a sharp constant, and contrast it with the known strict global inequality in higher dimensions. A natural two-bubble test family $u_\beta=v_\beta+v_{-\beta}$ is developed to probe minimizers, yielding explicit asymptotics and providing an alternative route to $c_{BE}(s)<c_{BE}^{\text{loc}}(s)$ for $d\ge2$, while numerical and exact computations in 1D with $p=3,4$ support the conjecture that no minimizer exists in $\dot{H}^s(\mathbb R)\setminus \mathcal B$. Collectively, the results illuminate a striking dimension-specific behavior of the BE inequality, connect bubble interactions with stability constants, and supply both analytic frameworks and numerical evidence for the 1D conjecture.

Abstract

In this paper, for $d \geq 1$ and $s \in (0,\frac{d}{2})$, we study the Bianchi-Egnell quotient \[ \mathcal Q(f) = \inf_{f \in \dot{H}^s(\mathbb R^d) \setminus \mathcal B} \frac{\|(-Δ)^{s/2} f\|_{L^2(\mathbb R^d)}^2 - S_{d,s} \|f\|_{L^{\frac{2d}{d-2s}}(\mathbb R^d)}^2}{\text{dist}_{\dot{H}^s(\mathbb R^d)}(f, \mathcal B)^2}, \qquad f \in \dot{H}^s(\mathbb R^d) \setminus \mathcal B, \] 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 $d \geq 2$ described recently in \cite{Koenig}. 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 $d \geq 2$. As a complement of the above, we study a family of test functions which interpolates between one and two Talenti bubbles, for every $d \geq 1$. For $d \geq 2$, this family yields an alternative proof of the main result of \cite{Koenig}. For $d =1$ we make some numerical observations which support the conjecture stated above.

Figures (5)

• Figure 4.1: Case $p = 3$: Plot of $y \mapsto e(y)$ for $d = 1$.
• Figure 4.2: Case $p = 3$: Plot of $y \mapsto e(y)$ for $d = 2$.
• Figure 4.3: Case $p = 3$: Plot of $y \mapsto e_1(y) - \frac{1}{5} e_2(y)$ for $d = 1$.
• Figure 4.4: Case $p = 4$: Plot of $y \mapsto f(y)$ for $d = 1$.
• Figure 4.5: Case $p = 4$: Plot of $y \mapsto g(y)$ for $d = 1$.

Theorems & Definitions (15)

• Theorem 1
• Conjecture 2
• Theorem 3
• Lemma 4
• proof
• proof : Proof of Theorem \ref{['theorem d=1 Sd']}; easy case $p \geq 4$
• proof : Proof of Theorem \ref{['theorem d=1 Sd']}, hard case $p < 4$
• Proposition 5
• proof : Proof of Proposition \ref{['proposition local two-bubbles']}
• Lemma 6