Projections in Sobolev spaces - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T05:45:44Z http://mathoverflow.net/feeds/question/75883 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/75883/projections-in-sobolev-spaces Projections in Sobolev spaces Radek Elor 2011-09-19T19:44:06Z 2011-09-19T23:48:56Z <p>I was browsing through the literature but I have not found anything related to my question:</p> <p>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?</p> <p>Any answers and hints will be appreciated. Of course, the question is interesting for \$p\neq 2\$.</p> http://mathoverflow.net/questions/75883/projections-in-sobolev-spaces/75894#75894 Answer by Bill Johnson for Projections in Sobolev spaces Bill Johnson 2011-09-19T23:48:56Z 2011-09-19T23:48:56Z <p>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 <code>\$W^{k,p}\$</code> 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.</p>