# On pointwise convergence of Schr\"odinger means

## TL;DR

This paper investigates the pointwise convergence of generalized Schr"odinger means for functions in Sobolev spaces, providing characterizations and sharp estimates for convergence almost everywhere, especially for decreasing convex sequences and the case a=1.

## Contribution

It offers a simple characterization of almost everywhere convergence for all functions in Sobolev spaces under specific conditions and establishes sharp estimates for maximal functions.

## Key findings

- Convergence characterized for 0<s<min{a/4,1/4} when a≠1.
- Sharp local and global estimates for maximal functions.
- Results include sharp convergence criteria for the case a=1.

## Abstract

For functions in the Sobolev space $H^s$ and decreasing sequences $t_n\to 0$ we examine convergence almost everywhere of the generalized Schr\"odinger means on the real line, given by \[S^af(x,t_n)=\exp( it_n (-\partial_{xx})^{a/2})f(x);\] here $a>0$, $a\neq 1$. For decreasing convex sequences we obtain a simple characterization of convergence a.e. for all functions in $H^s$ when $0<s<\min\{a/4,1/4\}$ and $a\neq 1$. We prove sharp quantitative local and global estimates for the associated maximal functions. We also obtain sharp results for the case $a=1$.