Pointwise Convergence for sequences of Schrödinger means in ${\mathbb{R}}^2$

Abstract

We consider pointwise convergence of Schrödinger means $e^{i{t}_n\Delta }f(x)$ for $f\in H^s({\mathbb{R}}^2)$ and decreasing sequences $\{t_n{\}}_{n=1}^{\infty }$ converging to zero. The main theorem improves the previous results of [Sjölin, JFAA, 2018] and [Sjölin-Strömberg, JMAA, 2020] in ${\mathbb{R}}^2$. This study is based on investigating properties of Schrödinger type maximal functions related to hypersurfaces with vanishing Gaussian curvature.