For functions in the Sobolev space Hs and decreasing sequences tn → 0 we examine convergence almost everywhere of the generalized Schrödinger means on the real line, given by Sf(x, tn) = exp(itn(−∂xx))f(x); here a > 0, a 6= 1. For decreasing convex sequences we obtain a simple characterization of convergence a.e. for all functions in Hs when 0 < s < min{a/4, 1/4} and a 6= 1. We prove sharp quantitative local and global estimates for the associated maximal functions. We also obtain sharp results for the case a = 1.

中文翻译：

---

关于薛定谔均值的点收敛

对于 Sobolev 空间 Hs 和递减序列 tn → 0 中的函数，我们几乎在实线上检查了广义薛定谔均值的收敛性，由 Sf(x, tn) = exp(itn(−∂xx))f(x) 给出; 这里 a > 0, a 6= 1。对于递减的凸序列，当 0 < s < min{a/4, 1/4} 和 a 6= 1 时，我们获得了 Hs 中所有函数的收敛 ae 的简单表征。我们证明相关最大函数的精确定量局部和全局估计。对于 a = 1 的情况，我们也获得了清晰的结果。