In this paper we show that the local Kato type smoothing estimates are essentially equivalent to the global Kato type smoothing estimates for some class of dispersive equations including the Schrödinger equation. From this we immediately have two results as follows. One is that the known local Kato smoothing estimates are sharp. The sharp regularity ranges of the global Kato smoothing estimates are already known, but those of the local Kato smoothing estimates are not. Sun et al. (Proc Am Math Soc 145(2):653–664, 2017) have shown it only in spacetime \(\mathbb R \times \mathbb R\). Our result resolves this issue in higher dimensions. The other one is the sharp global-in-time maximal Schrödinger estimates. Recently, the pointwise convergence conjecture of the Schrödinger equation has been settled by Du et al. (Ann Math 186:607–640, 2017) and Du and Zhang (Ann Math 189:837–861, 2019). For this they proved related sharp local-in-time maximal Schrödinger estimates. By our result, these lead to the sharp global-in-time maximal Schrödinger estimates.

中文翻译：

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色散方程的局部局部全局Kato型平滑估计

在本文中，我们表明，对于某些包括Schrödinger方程在内的色散方程，局部Kato型平滑估计基本上等于全局Kato型平滑估计。由此，我们立即得到两个结果，如下所示。一种是已知的本地Kato平滑估计很精确。全局Kato平滑估计的尖锐正则范围是已知的，但是本地Kato平滑估计的尖锐正则范围却未知。Sun等。（Proc Am Math Soc 145（2）：653–664，2017）仅在时空\（\ mathbb R \ times \ mathbb R \）中显示了它。我们的结果从更高的角度解决了这个问题。另一个是全球最大的薛定ding估计值。最近，Du等人解决了薛定ding方程的逐点收敛猜想。（Ann Math 186：607–640，2017）和Du and Zhang（Ann Math 189：837–861，2019）。为此，他们证明了相关的尖锐的局部时间最大Schrödinger估计。根据我们的结果，这些导致了Schrödinger全局最大时间的精确估计。