# Regularity of superposition operator generated by function between Banach spaces

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 $$K\to D$$.