Stability inequalities with explicit constants for a family of reverse Sobolev inequalities on the sphere

TL;DR

The paper proves a sharp stability inequality for a family of reverse Sobolev inequalities on the sphere across the full admissible regime $s-\frac{n}{2}\in(0,1)\cup(1,2)$. It overcomes the non-positivity of the operator $A_{2s}$ by introducing a modified distance $\mathsf d(u)$ to the optimizer manifold $\mathcal M$ and adapting Bianchi–Egnell’s method, aided by a center-of-mass balance argument and conformal invariance. The main results yield a positive stability constant $c_{BE}(s)$, with explicit values $c_{BE}(s)\le\frac{4s}{n+2s+2}$ for $s-\frac{n}{2}\in(0,1)$ and $c_{BE}(s)=1$ (not attained) for $s-\frac{n}{2}\in(1,2)$, providing the first explicit sharp constant in a Sobolev-type stability inequality and revealing non-attainment in the latter regime. The work also develops a local implicit-function framework around the optimizer manifold to support the stability analysis and discusses the existence of stability minimizers, though full existence is left open for certain parameter ranges. Overall, the paper advances the understanding of quantitative stability in reverse Sobolev inequalities on the sphere and offers techniques potentially applicable to related conformally invariant problems.

Abstract

We prove a stability inequality associated to the reverse Sobolev inequality on the sphere $\mathbb S^n$, for the full admissible parameter range $s - \frac{n}{2} \in (0,1) \cup (1,2)$. To implement the classical proof of Bianchi and Egnell, we overcome the main difficulty that the underlying operator $A_{2s}$ is not positive definite. As a consequence of our analysis and recent results from Gong et al. (arXiv:2503.20350 [math.AP]), the case $s - \frac{n}{2} \in (1,2)$ remarkably constitutes the first example of a Sobolev-type stability inequality (i) whose best constant is explicit and (ii) which does not admit an optimizer.

Theorems & Definitions (25)

• Theorem 1.2
• Theorem 1.3
• Proposition 1.4
• Proposition 2.1
• Lemma 2.2
• proof : Proof of Proposition \ref{['proposition balance condition']}
• proof : Proof of Lemma \ref{['lemma balance condition']}
• Lemma 2.3
• proof
• Lemma 2.4