  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.
