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For certain weighted locally convex spaces $X$ and $Y$ of one real variablesmooth functions, we characterize the smooth functions $\varphi: \mathbb{R} \to\mathbb{R}$ for which the composition operator $C_\varphi: X \to Y, \, f\mapsto f \circ \varphi$ is well-defined and continuous. This problem has beenrecently considered for $X = Y$ being the space $\mathscr{S}$ of rapidlydecreasing smooth functions [1] and the space $\mathscr{O}_M$ of slowlyincreasing smooth functions [2]. In particular, we recover both these resultsas well as obtain a characterization for $X =Y$ being the space $\mathscr{O}_C$of very slowly increasing smooth functions.

A note on composition operators between weighted spaces of smooth functions