Let $E$, $F$ be Banach spaces, $D$ be open in $E$, and $K=[0,1]$. Given $\varphi\colon K\times D\to F$ I call
$$
\varphi^\sharp\colon D^K\to F^K,\quad u\mapsto \varphi(\cdot,u(\cdot))
$$
the *superposition operator*.

I am interested in a run-down on how $\varphi^\sharp$ as a mapping between some vector spaces $V\subseteq D^K$ and $W\subseteq F^K$, both endowed with appropriate norms, inherits regularity from $\varphi$.

E.g., I can show that $\varphi^\sharp \in C^k\left(C\left(K,D\right),C\left(K,F\right)\right)%\tag{$\star$}$ provided $\varphi$ and $\partial_2^k\varphi$ are continuous.

I am especially interested in finding $V$ to be some Hilbert space that contains all piecewise smooth functions $K\to D$.

Many thanks in advance!