On localization of the Schr\"odinger maximal operator

TL;DR

This paper provides an alternative proof of the localization property of the Schr"odinger maximal operator using stationary phase, and explores implications for local and global inequalities and pointwise convergence.

Contribution

It offers a new proof technique for known localization results and extends understanding of Schr"odinger maximal inequalities through stationary phase analysis.

Findings

Equivalence of local and global Schr"odinger maximal inequalities.

Local inequality holds for functions in H^{3/8+}.

Almost everywhere convergence for functions in H^{3/8+}.

Abstract

In \cite{Lee:2006:schrod-converg}, when the spatial variable $x$ is localized, Lee observed that the Schr\"odinger maximal operator $e^{it\Delta}f(x)$ 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 $f\in H^{3/8+}$, which implies that $e^{it\Delta}f$ converges to $f$ almost everywhere if $f\in H^{3/8+}$. 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.