I am trying to prove the estimate $||F \circ u||_{H^k}\leq C_k||u||_{L^\infty}(1+||u||_{H^k})$, where $F$ is a smooth function with $F(0)=0$ and $u\in H_k\cap L_ \infty$.
It suffices to show $||\partial^\alpha (F\circ u)||_{L^2}\leq C_k||u||_{L^\infty}(1+||u||_{H^k})$ by definition of $||\cdot||_{H^k}$ for any multi-index $|\alpha|\leq k$
Let $l:=|\alpha|$, by chain rule, $\partial ^\alpha(F\circ u)=\sum_{\substack \beta_1+...+\beta_l=\alpha \\ 0 \leq i \leq l} C_{\beta_1,...,\beta_n,i}u^{\beta_{1}}...u^{\beta_l}(F^{(i)}\circ u)$.
Therefore, it suffices to show \begin{equation} ||u^{\beta_{1}}...u^{\beta_l}(F^{(i)} \circ u)||_{L^2}\lesssim ||u||_{L^\infty}(1+||u||_{H^k}) \end{equation}
By Holder's inequality, we have
\begin{align} ||u^{\beta_{1}}...u^{\beta_l}(F^{(i)}\circ u)||_{L^2} &\leq ||u^{\beta_{1}}...u^{\beta_l}||_{L^2}||||F^{(i)}\circ u||_{L^\infty}\\ &\lesssim||u||_{L^\infty}||u^{\beta_{1}}...u^{\beta_l}||_{L^2}\hspace{1cm}\text{(Mean-value Theorem)}\\ \end{align}
Now the work left is to estimate $||u^{\beta_{1}}...u^{\beta_l}||_{L^2}$.
\begin{align} ||u^{\beta_{1}}...u^{\beta_l}||_{L^2} &\leq||u^{\beta_1}||_{L_\frac{2l}{|\beta_ 1|}}...||u^{\beta_l}||_{L_\frac{2l}{|\beta_l|}} \hspace{1cm}\text{(Holder's Inequality)}\\ &\leq ||u||_{L^\infty}^{1-\frac{|\beta_1|}{l}} ||\partial^l u||_{L^2}^{\frac{|\beta|_1}{l}} ... ||u||_{L^\infty}^{1-\frac{|\beta_l|}{l}} ||\partial^l u||_{L^2}^{\frac{|\beta_l|}{l}} \hspace{1cm} \text{(Interpolation)}\\ &=||u||_{L^\infty}^{l-1}||\partial ^l u||_{L^2} \hspace{1cm}\text{(Grouping Terms)}\\ &\leq ||u||_{L^\infty}^{l-1}||u||_{H^k} \end{align}.
Now summing up my work, I can only show \begin{equation} ||u^{\beta_{1}}...u^{\beta_l}(F^{(i)} \circ u)||_{L^2}\lesssim ||u||_{L^\infty}^l||u||_{H^k} \end{equation}.
Any help would be appreciated.