Almost everywhere convergence on the solution of Schrödinger equation is an important problem raised by Carleson in harmonic analysis. In recent years, this problem was essentially solved by building the sharp $L^p$-estimate of Schrödinger maximal function. Du-Guth-Li in [8] proved the sharp $L^p$-estimates for all $p\ge 2$ in ${\mathbb{R}}^{2+1}$. Du-Zhang in [12] proved the sharp $L^2$-estimate in ${\mathbb{R}}^{n+1}$ with $n\ge 3$, but for $p>2$ the sharp $L^p$-estimate of Schrödinger maximal function is still unknown. In this paper, we obtain partial results on this problem by using polynomial partitioning and refined Strichartz estimates.

Schrödinger方程解的几乎所有地方的收敛都是Carleson在谐波分析中提出的重要问题。近年来，这个问题基本上是通过建立尖锐的${\mathrm{大号}}^p$-Schrödinger最大函数的估计。[8]中的Du-Guth-Li证明了敏锐的${\mathrm{大号}}^p$-估计所有 $p\ge \mathrm{2个}$ 在 ${\mathbb{[R}}^{\mathrm{2个}+\mathrm{1个}}$。[12]中的Du-Zhang证明了${\mathrm{大号}}^{\mathrm{2个}}$-估计 ${\mathbb{[R}}^{ñ+\mathrm{1个}}$ 和 $ñ\ge 3$， 但对于 $p>\mathrm{2个}$ 锋利的 ${\mathrm{大号}}^p$Schrödinger最大函数的估计仍然未知。在本文中，我们通过使用多项式划分和改进的Strichartz估计获得了关于此问题的部分结果。