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 $V\in L^{n/2}$.

中文翻译：

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包含奇异势的紧流形的统一 Sobolev 估计

我们获得了 Kenig、Ruiz 和第四作者 (1986) 对欧几里得空间和 Dos Santos Ferreira、Kenig 和 Salo (2014) 的一致 Sobolev 不等式的推广，用于涉及临界奇异势 $V\in L^{n/ 的紧黎曼流形2}$。我们还获得了第一、第三和第四作者 (2021)、Hassell 和 Tacy (2015)、第一和第四作者 (2019) 和 Hickman (2020) 的类似改进准模态估计，以及改进的统一模型的类似物Sobolev 对 Bourgain、Shao、第四作者和 Yao (2015) 和 Hickman (2020) 的估计，涉及此类潜力。此外，在 $S^n$ 上，我们获得了尖锐的均匀 Sobolev 不等式，其中涉及到最佳指数范围的这种势，这扩展了 S. Huang 和第四作者 (2014) 的结果。对于一般黎曼流形，