I have this inequality which I don't see how to prove it.
We have $F \in C^s$, and $u\in H^s$.
I want to show that:
$$\| F\circ u \|_{H^s} \leq C(\| F \circ u \|_{L^2}+\sum_{r=1}^s \sum_{j=1}^r \sum_{|a_1|+\ldots + |a_j| =r } \| F^{(j)}(u) \|_{L^{\infty}}\| D^{a_1}u \ldots D^{a_j}u \|_{L^2})$$
I tried through estimating the equivalent norm, i.e $\| F(u) \|_{H^s} = \| (1+|y|^s)\hat{F}(u) \|_{L^2}$, but didn't get me far, also tried the definition of $H^s$ norm, but didn't get me far.
Any hints or advice are welcomed. Edit: I added the constant C.