Pointwise convergence of sequential Schrödinger means

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 supn|eitn(−Δ)α/2f| can be deduced from those on sup0<t≤1|eit(−Δ)α/2f|, when { t_n} is contained in the Lorentz space ℓ^r,∞. 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.