Pointwise Convergence of the Fractional Schrödinger Equation in $\mathbb{R}^2$

Abstract

We investigate the pointwise convergence of the solution to the fractional Schrödinger 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 [19]. Our estimates extend the refined Strichartz estimates obtained by Du, Guth, and Li [10] to a general class of elliptic functions.