Convergence problem of Schrödinger equation in Fourier-Lebesgue spaces with rough data and random data

Abstract

In this paper, we consider the convergence problem of Schrödinger equation. Firstly, we show the almost everywhere pointwise convergence of Schrödinger equation in Fourier-Lebesgue spaces H ^ 1 p , p 2 ( R ) ( 4 ≤ p > ∞ ) , \hat {H}^{\frac {1}{p},\frac {p}{2}}(\mathbf {R})(4\leq p>\infty ), H ^ 3 s 1 p , 2 p 3 ( R 2 ) ( s 1 > 1 3 , 3 ≤ p > ∞ ) , \hat {H}^{\frac {3 s_{1}}{p},\frac {2p}{3}}(\mathbf {R}^{2})(s_{1}>\frac {1}{3},3\leq p>\infty ), H ^ 2 s 2 p , p ( R n ) ( s 2 > n 2 ( n + 1 ) , 2 ≤ p > ∞ , n ≥ 3 ) \hat {H}^{\frac {2 s_{2}}{p},p}(\mathbf {R}^{n})(s_{2}>\frac {n}{2(n+1)},2\leq p>\infty ,n\geq 3) with rough data. Secondly, we show that the maximal function estimate related to one dimensional Schrödinger equation can fail with data in H ^ s , p 2 ( R ) ( s > 1 p ) \hat {H}^{s,\frac {p}{2}}(\mathbf {R})(s>\frac {1}{p}) . Finally, we show the stochastic continuity of Schrödinger equation with random data in L ^ r ( R n ) ( 2 ≤ r > ∞ ) \hat {L}^{r}(\mathbf {R}^{n})(2\leq r>\infty ) almost surely. The main ingredients are maximal function estimates and density theorem in Fourier-Lebesgue spaces as well as some large deviation estimates.