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

Yan, Xiangqian; Zhao, Yajuan; Yan, Wei

doi:10.1090/proc/15841

Convergence problem of Schrödinger equation in Fourier-Lebesgue spaces with rough data and random data
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by Xiangqian Yan, Yajuan Zhao and Wei Yan PDF

Proc. Amer. Math. Soc. 150 (2022), 2455-2467 Request permission

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

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