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 Schrodinger 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 RxR. Our result resolves this issue in higher dimensions. The other one is the sharp global-in-time maximal Schrodinger estimates. Recently, the pointwise convergence conjecture of the Schrodinger 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 Schrodinger estimates. By our result, these lead to the sharp global-in-time maximal Schrodinger estimates