Power boundedness and related properties for weighted composition operators on $\mathscr{S}(\mathbb{R}^d)

We characterize those pairs $(\psi,\varphi)$ of smooth mappings $\psi:\mathbb{R}^d\rightarrow\mathbb{C},\varphi:\mathbb{R}^d\rightarrow\mathbb{R}^d$ for which the corresponding weighted composition operator $C_{\psi,\varphi}f=\psi\cdot(f\circ\varphi)$ acts continuously on $\mathscr{S}(\mathbb{R}^d)$. Additionally, we give several easy-to-check necessary and sufficient conditions of this property for interesting special cases. Moreover, we characterize power boundedness and topologizablity of $C_{\psi,\varphi}$ on $\mathscr{S}(\mathbb{R}^d)$ in terms of $\psi,\varphi$. Among other things, as an application of our results we show that for a univariate polynomial $\varphi$ with $\text{deg}(\varphi)\geq 2$, power boundedness of $C_{\psi,\varphi}$ on $\mathscr{S}(\mathbb{R})$ for every $\psi\in\mathscr{O}_M(\mathbb{R})$ only depends on $\varphi$ and that in this case power boundedness of $C_{\psi,\varphi}$ is equivalent to $(C_{\psi,\varphi}^n)_{n\in\mathbb{N}}$ converging to $0$ in $\mathcal{L}_b(\mathscr{S}(\mathbb{R}))$ as well as to the uniform mean ergodicity of $C_{\psi,\varphi}$. Additionally, we give an example of a power bounded and uniformly mean ergodic weighted composition operator $C_{\psi,\varphi}$ on $\mathscr{S}(\mathbb{R})$ for which neither the multiplication operator $f\mapsto \psi f$ nor the composition operator $f\mapsto f\circ\varphi$ acts on $\mathscr{S}(\mathbb{R})$. Our results complement and considerably extend various results of Fern\'andez, Galbis, and the second named author.