Pointwise convergence of sequential Schrödinger means

Abstract

We study pointwise convergence of the fractional Schrödinger means along sequences $t_n$ which converge to zero. Our main result is that bounds on the maximal function ${\sup }_n|e^{i{t}_n(-\Delta {)}^{\alpha /2}}f|$ 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.