Pointwise convergence of sequential Schrödinger means

Abstract

We study pointwise convergence of the fractional Schrödinger means along sequences tn 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 {tn} is contained in the Lorentz space ℓr,∞ . 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.