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

TL;DR

This paper advances the understanding of Schr"odinger maximal functions by establishing partial $L^p$-estimates in higher dimensions using polynomial partitioning, addressing a longstanding open problem in harmonic analysis.

Contribution

It provides new partial results on the $L^p$-estimates of Schr"odinger maximal functions in higher dimensions for $p>2$, employing polynomial partitioning techniques.

Findings

Established partial $L^p$-estimates for Schr"odinger maximal functions in higher dimensions.

Extended the range of $p$ for which sharp estimates are known beyond $p=2$.

Demonstrated the effectiveness of polynomial partitioning in this context.

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.