Let $E$, $F$ be Banach spaces, $D$ be open in $E$, and $K=[0,1]$. Given $\varphi\colon K\times D\to F$ $$ \varphi^\sharp\colon D^K\to F^K,\quad u\mapsto \varphi(\cdot,u(\cdot)) $$ is known as the superposition operator.
I am interested in the regularity of $\varphi^\sharp$ as a function on $C(K,D)$ where $C(K,D)$ is equipped with different $p$-norms and $\varphi$ is assumed to be sufficiently regular.
E.g., I can show that $$ \varphi^\sharp \in C^k\left(\left(C\left(K,D\right),\Vert\cdot\Vert_p\right),\left(C\left(K,F\right),\Vert\cdot\Vert_q\right)\right)\tag{$\star$} $$
if $p=q=\infty$, $k\in\mathbb{N}_0$, and $\varphi$ and $\partial^k_2\varphi$ are continuous. Can someone comment on the validity of $(\star)$ in dependence of $\varphi$ and $k$ for other values of $p$ and $q$? Any resources?
Thanks a lot in advance!