On the composition operators on Besov and Triebel–Lizorkin spaces with power weights

Let $G:\mathbb R\rightarrow \mathbb R$ be a continuous function. Under some assumptions on $G$, $s,\alpha ,p$ and $q$ we prove that $$\{G(f):f\in A_{p,q}^{s}(\mathbb R^{n},|\cdot |^{\alpha })\}\subset A_{p,q}^{s}(\mathbb R^{n},|\cdot |^{\alpha })$$ implies that $G$ is a linear function. Here $A_{p,q}^{s}(\mathbb R^{n},|\cdot |^{\alpha })$ stands either for the Besov space $B_{p,q}^{s}(\mathbb R^{n},|\cdot |^{\alpha })$ or for 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 with power weights.