CONVERGENCE PROBLEM OF SCHRODINGER EQUATION IN FOURIER-LEBESGUE SPACES WITH ROUGH DATA AND RANDOM DATA

作者：Yan, Xiangqian^*; Zhao, Yajuan; Yan, Wei

来源：PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY, 2022, 150(6): 2455-2467.

DOI：10.1090/proc/15841

• Convergence problem
• Schrodinger equation
• Fourier-Lebesgue spaces

In this paper, we consider the convergence problem of Schrodinger equation. Firstly, we show the almost everywhere pointwise convergence of Schrodinger equation in Fourier-Lebesgue spaces (H) over cap (1/p, p/2) (R)(4 <= p < infinity), <(H)over cap>(3s1/p, 2p/3) (R-2)(s(1) > 1/3, 3 <= p < infinity), <(H)over cap>(2s2/p,p) (R-n)(s(2) > n/2(n+1), 2 <= p < infinity, n >= 3) with rough data. Secondly, we show that the maximal function estimate related to one dimensional Schrodinger equation can fail with data in <(H)over cap>(s, p/2) (R)(s < 1/p). Finally, we show the stochastic continuity of Schrodinger equation with random data in <(L)over cap>(r) (R-n)(2 <= r < infinity) almost surely. The main ingredients are maximal function estimates and density theorem in Fourier-Lebesgue spaces as well as some large deviation estimates.