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

TL;DR

This paper explores the peculiar properties of Sobolev spaces for small p (0<p<1), revealing their isomorphism to L^p and discussing the implications of their trivial duals and pathological nature.

Contribution

It provides a detailed analysis of the behavior of Sobolev spaces when 0<p<1, expanding on Peetre's observations and clarifying their mathematical properties.

Findings

Sobolev spaces W^{k,p} are isomorphic to L^{p} for 0<p<1

The dual of W^{k,p} is trivial in this range

These spaces exhibit highly pathological features

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.