Stability of spinorial Sobolev inequalities on $\mathbb{S}^n$

TL;DR

This paper refines the spinorial Sobolev inequality on the sphere by establishing a stability version and investigates the properties of its extremizers, revealing they are not the optimizers of a related inequality.

Contribution

The paper introduces a stability inequality for the spinorial Sobolev inequality and analyzes the nature of extremizers, showing they are not optimal for another related inequality.

Findings

Established a stability inequality with an explicit constant.

Proved extremizers are not optimizers of a related Sobolev inequality.

Revealed the index and nullity of the extremizers.

Abstract

The spinorial Sobolev inequality on the unit sphere states \begin{equation*} \Big(\int| D\psi|^{\frac{2n}{n+1}}\Big)^{\frac{n+1}{n}}-\frac{n}{2}\omega_{n}^{1/n}\int\langle D\psi,\psi\rangle \geq 0, \end{equation*} with equality if and only if $\psi \in {\mathcal M}$, the set of all $-\frac 12$-Killing spinors and their conformal transformations. Our main result in this paper is to refine this inequality by establishing a stability inequality \begin{equation*} \Big(\int| D\psi|^{\frac{2n}{n+1}}\Big)^{\frac{n+1}{n}}-\frac{n}{2}\omega_{n}^{1/n}\int\langle D\psi,\psi\rangle \geq {\bf c}_S\inf_{\phi\in\mathcal{M}}\Big(\int| D(\psi-\phi)|^{\frac{2n}{n+1}}\Big)^{\frac{n+1}{n}}. \end{equation*} As a by-product of our argument, we show that elements in set $\mathcal M$ are not optimizers of another spinorial Sobolev inequality \begin{equation*} \Big(\int| D\psi|^{\frac{2n}{n+1}}\Big)^{\frac{n+1}{n}} \geq C_S \Big(\int|\psi|^{\frac{2n}{n-1}}\Big)^{\frac{n-1}{n}}, \end{equation*} unlike expected by experts. They have in fact index $n+1$ and nullity $2^{[\frac n2]+2}$.