Local $L^p$ norms of Schrödinger eigenfunctions on $\mathbb{S}^2$

On the canonical $2$-sphere and for Schrodinger eigenfunctions, we obtain a simple geometric criterion on the potential under which we can improve, near a given point and for every $p\neq 6$, Sogge's estimates by a power of the eigenvalue. This criterion can be formulated in terms of the critical points of the Radon transform of the potential and it is independent of the choice of eigenfunctions.