  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.
