Pointwise convergence of the fractional Schr\"odinger equation in ${\mathbb{R}}^2$

We investigate the pointwise convergence of the solution to the fractional Schr"odinger equation in ${\mathbb{R}}^2$. By establishing $H^s(\mathbb R^2)-L^3(\mathbb R^2)$ estimates for the associated maximal operator provided that $s>1\mathrm{/}3$, we improve the previous result obtained by Miao, Yang, and Zheng. Our estimates extend the refined Strichartz estimates obtained by Du, Guth, and Li to a general class of elliptic functions.