We prove variation-norm estimates for certain oscillatory integrals related to Carleson's theorem. Bounds for the corresponding maximal operators were first proven by Stein and Wainger. Our estimates are sharp in the range of exponents, up to endpoints. Such variation-norm estimates have applications to discrete analogues and ergodic theory. The proof relies on square function estimates for Schr\"odinger-like equations due to Lee, Rogers and Seeger. In dimension one, our proof additionally relies on a local smoothing estimate. Though the known endpoint local smoothing estimate by Rogers and Seeger is more than sufficient for our purpose, we also give a proof of certain local smoothing estimates using Bourgain-Guth iteration and the Bourgain-Demeter $\ell^2$ decoupling theorem. This may be of independent interest, because it improves the previously known range of exponents for spatial dimensions $n \ge 4$.

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与卡莱森定理相关的振荡积分的锐变范数估计

我们证明了与卡莱森定理相关的某些振荡积分的变异范数估计。Stein 和 Wainger 首先证明了相应极大算子的界限。我们的估计在指数范围内是尖锐的，直到端点。这种变异范数估计适用于离散类似物和遍历理论。该证明依赖于 Lee、Rogers 和 Seeger 的类 Schr\"odinger 方程的平方函数估计。在第一维中，我们的证明还依赖于局部平滑估计。虽然 Rogers 和 Seeger 的已知端点局部平滑估计更多对于我们的目的来说已经足够了，我们还使用 Bourgain-Guth 迭代和 Bourgain-Demeter $\ell^2$ 解耦定理给出了某些局部平滑估计的证明。这可能是独立的兴趣，