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 sup_n |e^it_n(-Δ )^α /2 f| can be deduced from those on sup_0< t≤ 1 |e^it(-Δ )^α /2 f| , 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.