Sobolev spaces in several variables in \(L^1\)-type norms are not isomorphic to Banach lattices (Q1811526)

Let \(k\) be a positive integer, let \(1<p<\infty\) and let (notation of the authors) \(L^p_{(k)} (\Omega)\) be the Sobolev space on an arbitrary non-empty open set \(\Omega\subset {\mathbb R}^n\), where \(n\geq 2\). The main result of the paper is the assertion that, under the above assumptions, the space \(L^1_{(k)} (\Omega)\) is uncomplemented in its second dual. As a corollary it is obtained that \(L^1_{(k)} (\Omega)\) is not isomorphic to any complemented subspace of any Banach lattice. The main tools for the proofs are a so-called Lindenstrauss lifting principle and a theorem due to J. Peetre on the non-existence of the right inverse for the Gagliardo trace. The paper is self-contained and clearly written.