Convergence problem of Schr\"odinger equation in Fourier-Lebesgue spaces with rough data and random data

TL;DR

This paper investigates the convergence properties of the Schr"odinger equation in Fourier-Lebesgue spaces with rough and random data, establishing almost everywhere convergence, limitations of maximal function estimates, and stochastic continuity results.

Contribution

It provides new convergence results for Schr"odinger equations in Fourier-Lebesgue spaces with rough data and demonstrates stochastic continuity with random data.

Findings

Almost everywhere convergence in specified Fourier-Lebesgue spaces.

Failure of maximal function estimates for certain data regularities.

Stochastic continuity of Schr"odinger equation with random data almost surely.

Abstract

In this paper, we consider the convergence problem of Schr\"odinger equation. Firstly, we show the almost everywhere pointwise convergence of Schr\"odinger equation in Fourier-Lebesgue spaces $\hat{H}^{\frac{1}{p},\frac{p}{2}}(\mathbb{R})(4\leq p<\infty),$ $\hat{H}^{\frac{3 s_{1}}{p},\frac{2p}{3}}(\mathbb{R}^2)(s_{1}>\frac{1}{3},3\leq p<\infty),$ $\hat{H}^{\frac{2 s_{1}}{p},p}(\mathbb{R}^n)(s_{1}>\frac{n}{2(n+1)},2\leq p<\infty,n\geq3)$ with rough data. Secondly, we show that the maximal function estimate related to one Schr\"odinger equation can fail with data in $\hat{H}^{s,\frac{p}{2}}(\mathbb{R})(s<\frac{1}{p})$. Finally, we show the stochastic continuity of Schr\"odinger equation with random data in $\hat{L}^{r}(\mathbb{R}^n)(2\leq r<\infty)$ almost surely. The main ingredients are Lemmas 2.4, 2.5, 3.2-3.4.