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

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 $s>3/8$.