论文标题
傅里叶 - lebesgue空间中schrödinger方程的收敛问题，带有粗糙的数据和随机数据
Convergence problem of Schrödinger equation in Fourier-Lebesgue spaces with rough data and random data
论文作者
论文摘要
在本文中，我们考虑了Schrödinger方程的收敛问题。首先，我们在傅立叶 - lebesgue Spaces中几乎可以显示Schrödinger方程的几乎偶然的融合， $ \ hat {h}^{\ frac {3 s_ {1}} {p}，\ frac {2p} {3}}} {3}}（\ Mathbb {r}^2）（s_ {1}> \ frac> \ frac {1} {1} {1} {3} {3} {3} {3 \ 3 \ leq p <\ feftty） $ \ hat {h}^{\ frac {2 s_ {1}} {p}，p}（\ mathbb {r}^n）（s_ {1}> \ frac {n} {n} {n} {2（n+1）}，2 \ leq p <leq p <\ iffty，n \ geq3）其次，我们表明，与一个schrödinger方程相关的最大函数估计可能会因$ \ hat {h}^{s，\ frac {p} {2}}}（\ mathbb {r}）（s <\ frac {1} {1} {p}）$而失败。最后，我们在$ \ hat {l}^{r}（\ mathbb {r}^n）（2 \ leq r <\ infty）$几乎肯定地显示了Schrödinger方程的随机连续性。主要成分是引理2.4、2.5、3.2-3.4。
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 $\hat{H}^{\frac{1}{p},\frac{p}{2}}(\mathbb{R})(4\leq p<\infty),$ $\hat{H}^{\frac{3 s_{1}}{p},\frac{2p}{3}}(\mathbb{R}^2)(s_{1}>\frac{1}{3},3\leq p<\infty),$ $\hat{H}^{\frac{2 s_{1}}{p},p}(\mathbb{R}^n)(s_{1}>\frac{n}{2(n+1)},2\leq p<\infty,n\geq3)$ with rough data. Secondly, we show that the maximal function estimate related to one Schrödinger 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 Schrödinger 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.
