Sharp convergence for sequences of Schrödinger means and related generalizations

TL;DR

This paper resolves the pointwise convergence problem for Schrödinger means along decreasing time sequences by establishing sharp almost everywhere convergence for $f\in H^s(\mathbb{R}^N)$ when $s>s_0$, with $s_0=\min\{ \frac{r}{\frac{N+1}{N}r+1}, \frac{N}{2(N+1)}\}$ and $t_n\in\ell^{r,\infty}$. The authors develop a frequency-localized framework combining Littlewood–Paley analysis, wave-packet decomposition, and orthogonality to derive a strong $L^2$ maximal estimate on balls, then extend the analysis to fractional and nonelliptic Schrödinger means. They further show endpoint sharpness via Luca–Rogers and RVV-type counterexamples, illustrating a nuanced dependence on the sparsity parameter $r$. The results yield a coherent picture of when discrete-time convergence improves upon the continuous-time thresholds, and they provide sharp tools for related Schrödinger-type operators with potential applications in dispersive analysis and PDEs. Overall, the paper advances our understanding of discrete-time evolution equations and sharp Sobolev-regularity requirements for almost everywhere convergence.

Abstract

For decreasing sequences $\{t_{n}\}_{n=1}^{\infty}$ converging to zero, we obtain the almost everywhere convergence results for sequences of Schrödinger means $e^{it_{n}Δ}f$, where $f \in H^{s}(\mathbb{R}^{N}), N\geq 2$. The convergence results are sharp up to the endpoints, and the method can also be applied to get the convergence results for the fractional Schrödinger means and nonelliptic Schrödinger means.