A note on composition operators between weighted spaces of smooth functions

For certain weighted locally convex spaces $X$ and $Y$ of one real variable smooth functions, we characterize the smooth functions $\varphi: $\mathbb{R}$ \to $\mathbb{R}$$forwhichthecompositionoperator$C_\varphi: X \to Y, , f \mapsto f \circ \varphi$ is well-defined and continuous. This problem has been recently considered for $X=Y$ being the space $\mathcal{S}$ of rapidly decreasing smooth functions [1] and the space ${\mathcal{O}}_M$ of slowly increasing smooth functions [2]. In particular, we recover both these results as well as obtain a characterization for $X=Y$ being the space ${\mathcal{O}}_C$ of very slowly increasing smooth functions.