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.
Categories: math.AP