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

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