Global Kato type smoothing estimates via local ones for dispersive equations

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 Schr\"odinger 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. Recently, Sun, Tr\'elat, Zhang and Zhong 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 Schr\"odinger estimates. Recently, the pointwise convergence conjecture of the Schr\"odinger equation has been settled by Du--Guth--Li--Zhang and Du--Zhang. For this they proved related sharp local maximal Schr\"odinger estimates. By our result, these lead to the sharp global-in-time maximal Schr\"odinger estimates.