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

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.