Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

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 $p\neq 2$.

share|improve this question
1  
Sorry for the stupid question: what does your last question mean? (Isomorphic to squares) –  András Bátkai Sep 19 '11 at 20:03
    
My guess is that the last question means: is $E$ isomorphic as a Banach space to $E\oplus E$? –  Yemon Choi Sep 20 '11 at 1:27
add comment

1 Answer

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.

share|improve this answer
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.