$L^p$-estimate of Schrödinger maximal function in higher dimensions

Abstract

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 \cite{DGL} proved the sharp $L^p$-estimates for all $p\ge 2$ in ${\mathbb{R}}^{2+1}$. Du-Zhang in \cite{DZ} 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.

26 pages, In this version, we made an appropriate modification to Theorem 1.1

References in corpus (3)

• Schrödinger maximal function estimates via the pseudoconformal transformation
• An improved necessary condition for the Schrödinger maximal estimate
• $L^p$-estimates of maximal function related to Schrödinger Equation in ${\mathbb{R}}^2$