$L^{p}$-estimate of Schr\"odinger maximal function in higher dimensions

Abstract

Almost everywhere convergence on the solution of Schr\"odinger 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 Schr\"odinger maximal function. Du-Guth-Li in \cite{DGL} proved the sharp $L^p$-estimates for all $p \geq 2$ in $\mathbb{R}^{2+1}$. Du-Zhang in \cite{DZ} proved the sharp $L^2$-estimate in $\mathbb{R}^{n+1}$ with $n \geq 3$, but for $p>2$ the sharp $L^p$-estimate of Schr\"odinger maximal function is still unknown. In this paper, we obtain partial results on this problem by using polynomial partitioning.