Uniform Sobolev estimates in $\mathbb{R}^{n}$ involving singular potentials

Abstract

We generalize the Stein-Tomas [17] $L^2$-restricition theorem and the uniform Sobolev estimates of Kenig, Ruiz and the second author [11] by allowing critically singular potential. We also obtain Strichartz estimates for Schr\"odinger and wave operators with such potentials. Due to the fact that there may be nontrivial eigenfunctions we are required to make certain spectral assumptions, such as assuming that the solutions only involve sufficiently large frequencies.