Pointwise convergence of sequential Schrödinger means

Abstract

We study pointwise convergence of the fractional Schrödinger means along sequences t_{n} that 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< tle 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 and Seeger, and Li, Wang, and 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.