Pointwise Convergence for sequences of Schr\"{o}dinger means in $\mathbb{R}^{2}$

TL;DR

This paper investigates the pointwise convergence of Schrödinger means in two-dimensional space for functions in Sobolev spaces, improving previous convergence results by analyzing maximal functions related to hypersurfaces with zero Gaussian curvature.

Contribution

It advances the understanding of Schrödinger mean convergence in 2D by refining conditions and employing properties of maximal functions associated with special hypersurfaces.

Findings

Improved convergence results for Schrödinger means in b2.

Established connections between maximal functions and hypersurfaces with vanishing Gaussian curvature.

Enhanced understanding of pointwise convergence in Sobolev spaces.

Abstract

We consider pointwise convergence of Schr\"{o}dinger means $e^{it_{n}\Delta}f(x)$ 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 [Sj\"{o}lin, JFAA, 2018] and [Sj\"{o}lin-Str\"{o}mberg, JMAA, 2020] in $\mathbb{R}^{2}$. This study is based on investigating properties of Schr\"{o}dinger type maximal functions related to hypersurfaces with vanishing Gaussian curvature.