Uniform Sobolev estimates on compact manifolds involving singular potentials

Abstract

We obtain generalizations of the uniform Sobolev inequalities of Kenig, Ruiz and the fourth author (1986) for Euclidean spaces and Dos Santos Ferreira, Kenig and Salo (2014) for compact Riemannian manifolds involving critically singular potentials $V\in L^{n/2}$. We also obtain the analogous improved quasimode estimates of the first, third and fourth author (2021), Hassell and Tacy (2015), the first and fourth author (2019), and Hickman (2020), as well as analogues of the improved uniform Sobolev estimates of Bourgain, Shao, the fourth author and Yao (2015), and Hickman (2020), involving such potentials. Additionally, on $S^n$, we obtain sharp uniform Sobolev inequalities involving such potentials for the optimal range of exponents, which extend the results of S. Huang and the fourth author (2014). For general Riemannian manifolds, we improve the earlier results in of the first, third and fourth authors (2021) by obtaining quasimode estimates for a larger (and optimal) range of exponents under the weaker assumption that $V\in L^{n/2}$.