Pointwise convergence of sequential Schrödinger means

Abstract

AbstractWe 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^{it_{n}(-\Delta )^{\alpha /2}} f| $ sup n | e i t n ( − Δ ) α / 2 f | can be deduced from those on $\sup_{0< t\le 1} |e^{it(-\Delta )^{\alpha /2}} f|$ sup 0 < t ≤ 1 | e i t ( − Δ ) α / 2 f | , when $\{t_{n}\}$ { t n } is contained in the Lorentz space $\ell ^{r,\infty}$ 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.