Pointwise convergence of sequential Schr\"odinger means

TL;DR

This paper investigates the pointwise convergence of fractional Schr"odinger means along sequences tending to zero, establishing bounds on maximal functions and extending recent results to higher dimensions using a localization approach.

Contribution

It provides new bounds on maximal functions for fractional Schr"odinger means along sequences in Lorentz spaces, extending convergence results to higher dimensions.

Findings

Bounds on maximal functions can be deduced from those on continuous parameters.

Results extend recent work to higher dimensions.

The localization approach applies to other dispersive equations.

Abstract

We study pointwise convergence of the fractional Schr\"odinger means along sequences $t_n$ which converge to zero. Our main result is that bounds on the maximal function $\sup_{n} |e^{it_n(-\Delta)^{\alpha/2}} f| $ can be deduced from those on $\sup_{0<t\le 1} |e^{it(-\Delta)^{\alpha/2}} f|$ when $\{t_n\}$ is contained in the Lorentz space $\ell^{r,\infty}$. Consequently, our results provide seemingly optimal results in higher dimensions, which extend the recent work of Dimou-Seeger, and Li-Wang-Yan to higher dimensions. Our approach based on a localization argument also works for other dispersive equations and provides alternative proofs of previous results on sequential convergence.