On localization of the Schr\"odinger maximal operator

In \cite{Lee:2006:schrod-converg}, when the spatial variable $x$ is localized, Lee observed that the Schr"odinger maximal operator $e^{it\mathrm{\Delta }}f(x)$ enjoys certain localization property in $t$ for frequency localized functions. In this note, we give an alternative proof of this observation by using the method of stationary phase, and then include two applications: the first is on is on the equivalence of the local and the global Schr"odinger maximal inequalities; secondly the local Schr"odinger maximal inequality holds for $f\in H^{3\mathrm{/}8+}$, which implies that $e^{it\mathrm{\Delta }}f$ converges to $f$ almost everywhere if $f\in H^{3\mathrm{/}8+}$. These results are not new. In this note we would like to explore them from a slightly different perspective, where the analysis of the stationary phase plays an important role.