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

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 ${\lim }_{t\to 0}{e}^{it\mathrm{\Delta }}f(x)=f(x)$ almost everywhere for all $f\in H^s({\mathbb{R}}^2)$ provided that $s>3\mathrm{/}8$.