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

الأصل · EN

We discuss a phenomenon observed by Jaak Peetre in the seventies: for small Lᵖ-exponents, i.e. for 0<p<1, the Sobolev spaces Wᵏ,ᵖ defined in a seemingly natural way are isomorphic to Lᵖ. This says that the dual of Wᵏ,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.