The rate of convergence on Schrödinger operator (Q669492)

In this paper, the authors discuss the rate of convergence on $e^{it\triangle}(f)$ for $f$ in the Sobolev space $H^s(\mathbb{R}^n)$. \par Theorem. If $s>1/3$ and $0 \leq \delta < 2$, then for all $f \in H^{s+\delta}(\mathbb{R}^2)$ $e^{it\triangle}(f)(x)-f(x)=o(t^{\delta/2})$ a. e. as $t \rightarrow 0$. \par Theorem. If $s \geq1/4$ \ $\alpha > 1$ and $0 \leq \delta < \alpha$, then for all $f \in H^{s+\delta}(\mathbb{R})$ $e^{it|\triangle|^{\alpha/2}}(f)(x)-f(x)=o(t^{\delta/\alpha})$ a. e. as $t \rightarrow 0$. \par Theorem. If $s>n/2(n+1)$ and $0 \leq \delta < 2$, then for all $f \in H^{s+\delta}(\mathbb{R}^n)$ $e^{it\triangle}(f)(x)-f(x)=o(t^{\delta/2})$ a. e. as $t \rightarrow 0$.