Let $ E $, $ F $ to be Banach spaces, $ D $ to be open in $ E $, and $ K =[0,1]$. Given $ varphi colon K times D to F $ J & # 39; calls

$$

varphi ^ sharp colon D ^ K to F ^ K, quad mapsto varphi ( cdot, u ( cdot))

$$

the *overlay operator*.

I am interested in an overview of how $ varphi ^ sharp $ as a mapping between vector spaces $ V subseteq D ^ K $ and $ W subseteq F ^ K $, both with appropriate standards, inherit regularity $ varphi $.

For example, 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 particularly interested in finding $ V $ to be a Hilbert space containing all the piecewise smooth functions K $ to D $.

Thanks in advance!