L-estimate of Schrödinger maximal function in higher dimensions

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