Frequent hypercyclicity of weighted composition operators on the space of smooth functions

We prove that every hypercyclic weighted composition operator acting on the space of smooth functions on the real line is already frequently hypercyclic. Moreover, for a given frequently hypercyclic weighted composition operator Cw,ψ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C_{w,\psi }$$\end{document} we show that C∞(R)=FHC(Cw,ψ)+FHC(Cw,ψ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C^\infty ({\mathbb {R}})=FHC(C_{w,\psi })+FHC(C_{w,\psi })$$\end{document} and that FHC(Cw,ψ)∪{0}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$FHC(C_{w,\psi })\cup \{0\}$$\end{document} contains a closed infinite dimensional subspace.