A note on composition operators between weighted spaces of smooth functions

TL;DR

This paper characterizes the functions  for which composition operators between certain weighted spaces of smooth functions are well-defined and continuous, extending previous results to new classes of function spaces.

Contribution

It provides a comprehensive characterization of composition operators between weighted spaces of smooth functions, including new classes like  of very slowly increasing functions.

Findings

Characterization of  functions for well-defined composition operators

Extension of previous results to  spaces

Unified framework for various weighted smooth function spaces

Abstract

For certain weighted locally convex spaces $X$ and $Y$ of one real variable smooth functions, we characterize the smooth functions $\varphi: \mathbb{R} \to \mathbb{R}$ for which the composition operator $C_\varphi: X \to Y, \, f \mapsto f \circ \varphi$ is well-defined and continuous. This problem has been recently considered for $X = Y$ being the space $\mathscr{S}$ of rapidly decreasing smooth functions [1] and the space $\mathscr{O}_M$ 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 $\mathscr{O}_C$ of very slowly increasing smooth functions.