$L^p$-estimates of maximal function related to Schr\"{o}dinger Equation in $\mathbb{R}^2$

TL;DR

This paper employs Guth's polynomial partitioning to derive $L^p$ bounds for the maximal function of Schr"odinger solutions in two dimensions, improving understanding of pointwise convergence for functions in certain Sobolev spaces.

Contribution

It introduces a novel application of polynomial partitioning to establish $L^p$ estimates for Schr"odinger maximal functions in $\

Findings

Established new $L^p$ bounds for Schr"odinger maximal functions in $\

Recovered almost everywhere convergence for initial data in $H^s(\

Improved previous results on pointwise convergence for $s > 3/8$.

Abstract

Using Guth's polynomial partitioning method, we obtain $L^p$ estimates for the maximal function associated to the solution of Schr\"odinger equation in $\mathbb R^2$. The $L^p$ estimates can be used to recover the previous best known result that $\lim_{t \to 0} e^{it\Delta}f(x)=f(x)$ almost everywhere for all $f \in H^s (\mathbb{R}^2)$ provided that $s>3/8$.