I was browsing through the literature but I have not found anything related to my question:

I am interested in decompositions of functions in Sobolev spaces $W^{k,p}(\Omega)$, where $\Omega$ is some region in $\mathbb{R}^n$. Can we list all (any non-trivial?) complemented subspaces of Sobolev spaces? Is there an easy argument to show that they are isomorphic to their squares?

Any answers and hints will be appreciated. Of course, the question is interesting for $p\neq 2$.