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

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.