Local $L^p$ norms of Schr{\"o}dinger eigenfunctions on $\mathbb{S}^2$

TL;DR

This paper establishes a geometric criterion involving the Radon transform of the potential that allows for improved local $L^p$ estimates of Schr{"o}dinger eigenfunctions on the 2-sphere, surpassing classical bounds for certain $p$.

Contribution

It introduces a novel geometric condition based on the Radon transform of the potential that enhances local $L^p$ bounds for Schr{"o}dinger eigenfunctions on $S^2$, independent of eigenfunction choice.

Findings

Improved local $L^p$ estimates for Schr{"o}dinger eigenfunctions under the new criterion.

The criterion depends on the critical points of the Radon transform of the potential.

Enhancement applies near a given point for all $p eq 6$, surpassing classical Sogge estimates.

Abstract

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.