Lp-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 Lp-estimate of Schrödinger maximal function. Du-Guth-Li in [8] proved the sharp Lp-estimates for all p≥2 in R2+1. Du-Zhang in [12] proved the sharp L2-estimate in Rn+1 with n≥3, but for p>2 the sharp Lp-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.