标题： 色散方程在Damek-Ricci空间上渐近凹相位的正则性和点态收敛性 显示英文标题

标题： Regularity and pointwise convergence for dispersive equations with asymptotically concave phase on Damek-Ricci spaces

摘要： 我们研究了Carleson问题在Damek-Ricci空间 $S$ 上色散方程的径向初值问题：\begin{equation*} \begin{cases} i\frac{\partial u}{\partial t} +\Psi(\sqrt{-\mathcal{L}} )u=0\:,\: (x,t) \in S \times \mathbb{R} \:, \\ u(0,\cdot)=f\:,\: \text{ on } S \:, \end{cases} \end{equation*}，其中 $\mathcal{L}= \Delta$ 是Laplace-Beltrami算子或 $\tilde{\Delta}$ 是移位的Laplace-Beltrami算子，使得相应的相函数 $\psi$ 对某个 $a \in (0,1)$，满足大频率渐近性：\begin{equation*} \psi(\lambda)=\lambda^a + \mathcal{O}(1)\:,\:\: \lambda \gg 1\:. \end{equation*}。对于解 $u$ 几乎处处点态收敛到其径向初值 $f$，我们得到了几乎最优的正则性阈值 $\beta>a/4$。 这一结果即使对于$\mathbb{R}^n$也是新的，并且在分数阶薛定谔方程的特殊情况下，推广了 Walther 的经典欧几里得结果。 显示英文摘要

摘要： We study the Carleson's problem on Damek-Ricci spaces $S$ for dispersive equations: \begin{equation*} \begin{cases} i\frac{\partial u}{\partial t} +\Psi(\sqrt{-\mathcal{L}} )u=0\:,\: (x,t) \in S \times \mathbb{R} \:, \\ u(0,\cdot)=f\:,\: \text{ on } S \:, \end{cases} \end{equation*} where $\mathcal{L}= \Delta$, the Laplace-Beltrami operator or $\tilde{\Delta}$, the shifted Laplace-Beltrami operator, so that the corresponding phase function $\psi$ satisfies for some $a \in (0,1)$, the large frequency asymptotic: \begin{equation*} \psi(\lambda)=\lambda^a + \mathcal{O}(1)\:,\:\: \lambda \gg 1\:. \end{equation*} For almost everywhere pointwise convergence of the solution $u$ to its radial initial data $f$, we obtain the almost sharp regularity threshold $\beta>a/4$. This result is new even for $\mathbb{R}^n$ and in the special case of the fractional Schr\"odinger equations, generalizes classical Euclidean results of Walther.