  On the canonical $2$-sphere and for Schr{\"o}dinger 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.
