  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 $s>1/3$, 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.
