On the composition operators on Besov and Triebel-Lizorkin spaces of power weights

Abstract

Let $G:\mathbb{R\rightarrow R}$ be a continuous function. Under some assumptions on $G$, $s,\alpha ,p$ and $q$ we prove that \begin{equation*} \{G(f):f\in A_{p,q}^{s}(\mathbb{R}^{n},|\cdot |^{\alpha })\}\subset A_{p,q}^{s}(\mathbb{R}^{n},|\cdot |^{\alpha }) \end{equation*} implies $G$ is a linear function. Here $A_{p,q}^{s}(\mathbb{R}^{n},|\cdot|^{\alpha })$ stands for either the Besov space $B_{p,q}^{s}(\mathbb{R}^{n},|\cdot |^{\alpha })$ or the Triebel-Lizorkin space $F_{p,q}^{s}(\mathbb{R}^{n},|\cdot |^{\alpha })$. These spaces unify and generalize many classical function spaces such as Sobolev spaces of power weights. One of the main difficulties to study this problem is that the norm of the $A_{p,q}^{s}(\mathbb{R}^{n},|\cdot |^{\alpha })$ spaces with $\alpha \neq 0$ is not translation invariant, so some new techniques must be developed.