Degenerate stability of some Sobolev inequalities

TL;DR

This paper investigates the stability of certain Sobolev inequalities on specific spheres, revealing a fourth power remainder term that is optimal, contrasting with typical second-order vanishing, and employs an iterative Bianchi-Egnell approach.

Contribution

It demonstrates a novel fourth power stability estimate for conformally invariant Sobolev inequalities on spheres, extending understanding of stability phenomena beyond traditional second-order results.

Findings

Optimal fourth power remainder term established

Stability behavior differs from classical second-order vanishing

Methodology involves an iterative Bianchi-Egnell strategy

Abstract

We show that on $\mathbb S^1(1/\sqrt{d-2})\times\mathbb S^{d-1}(1)$ the conformally invariant Sobolev inequality holds with a remainder term that is the fourth power of the distance to the optimizers. The fourth power is best possible. This is in contrast to the more usual vanishing to second order and is motivated by work of Engelstein, Neumayer and Spolaor. A similar phenomenon arises for subcritical Sobolev inequalities on $\mathbb S^d$. Our proof proceeds by an iterated Bianchi-Egnell strategy.