Stability of spinorial Sobolev inequalities on $\mathbb{S}^n$
The spinorial Sobolev inequality on the unit sphere states \begin{equation*} \Big(\int| Dψ|^{\frac{2n}{n+1}}\Big)^{\frac{n+1}{n}}-\frac{n}{2}ω_{n}^{1/n}\int\langle Dψ,ψ\rangle \geq 0, \end{equation*} with equality if and only if $ψ\in {\mathcal M}$, 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| Dψ|^{\frac{2n}{n+1}}\Big)^{\frac{n+1}{n}}-\frac{n}{2}ω_{n}^{1/n}\int\langle Dψ,ψ\rangle \geq {\bf c}_S\inf_{ϕ\in\mathcal{M}}\Big(\int| D(ψ-ϕ)|^{\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| Dψ|^{\frac{2n}{n+1}}\Big)^{\frac{n+1}{n}} \geq C_S \Big(\int|ψ|^{\frac{2n}{n-1}}\Big)^{\frac{n-1}{n}}, \end{equation*} unlike expected by experts. They have in fact index $n+1$ and nullity $2^{[\frac n2]+2}$.