A Further Remark on Sobolev Spaces. The Case $0<p<1$

Abstract

We discuss a phenomenon observed by Jaak Peetre in the seventies: for small $L^p$-exponents, i.e. for $0<p<1$, the Sobolev spaces $W^{k,p}$ defined in a seemingly natural way are isomorphic to $L^p$. This says that the dual of $W^{k,p}$ is trivial, and indicates that these spaces are highly pathological. In this note we expand on Peetre's observation, explaining in detail some points that might merit further discussion.

arXiv comment5 pages, corrected typos