We know that if $G\in \mathcal{C}^1(\mathbb{R})$ such that $G(0)=0$ and if $u \in W^{1,p}(I)$, then $G(u) \in W^{1,p}(I)$ and $(G(u))' = G'(u) u'$.
Do we have a similar result for $W^{s,p}(I)$ with $s>1$ ? Or for the particular case $W^{s,p}(I) = H^s(\mathbb{T})$ ?
In fact, I tried to look for one, at least for the particular case where $u \in H^s(\mathbb{T})$: We have $$ \|G(u)\|_s^2:= \sum_{n\in \mathbb{Z}}(1+n^2)^s |\widehat{G(u)}(n)|^2.$$ The question is: which assumption we can give to $G$, so that we can compare $|\widehat{G(u)}|$ with $|\hat{u}|$.
Thank you in advanced