$L^{P} ( \Omega ; W^{1,P}(Y))$ can someone define this space. $L^{P}$ and $W^{1,P}$ are sobolve and Lp spaces. How norm can define? I have this space $L^{P} ( \Omega ; W^{1,P}_{per}(Y))$ can someone define it. As far I understands it takes a function f from $L^{P}(\Omega)$ and maps it to Sobolov space with periodicity $Y$.
Or $f : W^{1,P}_{per}(Y) \rightarrow L^{P}(\Omega)$.
i am confused here???
here $L^{P}$ and $W^{1,P}$ are sobolov and Lp spaces. Y is the periodicity of domain $\Omega$.
How norm can define?
$||f||_{L^{P} ( \Omega ; W^{1,P}_{per}(Y))} = \int_{\Omega}  ||f||_{W^{1,P}_{per}(Y)}  dx$

Functions in $L^P(\Omega; W_{per}^{1, P}(Y))$ are functions $f: \Omega \rightarrow W_{per}^{1, P}(Y)$ s.t. the norm on the space is well-defined at $f$. A suitable norm is
$$
\lVert f \rVert_{L^P(\Omega; W_{per}^{1, P}(Y))} := \left (\int_\Omega \lVert f(x) \rVert_{W_{per}^{1, P}(Y)}^p ~\mathrm{d}x \right)^\frac{1}{p}.
$$
