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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 $p\neq 2$.

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    $\begingroup$ Sorry for the stupid question: what does your last question mean? (Isomorphic to squares) $\endgroup$
    – András Bátkai
    Commented Sep 19, 2011 at 20:03
  • $\begingroup$ My guess is that the last question means: is $E$ isomorphic as a Banach space to $E\oplus E$? $\endgroup$
    – Yemon Choi
    Commented Sep 20, 2011 at 1:27

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Read the article by Pelczynski and Wojciechowski in vol. 2 of the Handbook of the Geometry of Banach Spaces (North-Holland). In the reflexive range, under mild conditions the space $W^{k,p}$ is isomorphic to $L_p$, so you are asking about the class of complemented subspaces of $L_p$, which has received a lot of attention in Banach space theory. Bourgain, Rosenthal, and Schechtman proved that there are uncountably many isomorphically different ones, but it is unknown if there are a continuum of them. It is not known whether all complemented subspaces of $L_p$ are isomorphic to their squares or whether each one has an unconditional basis (they all do have Schauder bases, however). See the article by Alspach and Odell in vol. 1 of the Handbook of the Geometry of Banach Spaces.

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