Pointwise convergence of sequential Schrödinger means

Abstract

Abstract We study pointwise convergence of the fractional Schrödinger means along sequences t n $t_{n}$ that converge to zero. Our main result is that bounds on the maximal function sup n | e i t n ( − Δ ) α / 2 f | $\sup_{n} |e^{it_{n}(-\Delta )^{\alpha /2}} f| $ can be deduced from those on sup 0 < t ≤ 1 | e i t ( − Δ ) α / 2 f | $\sup_{0< t\le 1} |e^{it(-\Delta )^{\alpha /2}} f|$ , when { t n } $\{t_{n}\}$ is contained in the Lorentz space ℓ r , ∞ $\ell ^{r,\infty}$ . 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.

Keywords

• Pointwise convergence
• Schrödinger operator