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.