A note on composition operators between weighted spaces of smooth functions

Abstract

For certain weighted locally convex spaces $X$ and $Y$ of one real variable smooth functions, we characterize the smooth functions $\phi :\mathbb{R}\to \mathbb{R}$ for which the composition operator $C_{\phi }:X\to Y,f\mapsto f\circ \phi$ 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.

9 pages