A note on homogeneous Sobolev spaces of fractional order

TL;DR

This paper investigates the differences between two definitions of homogeneous fractional Sobolev spaces, revealing that they do not always coincide and exploring conditions under which they do, along with some peculiar behaviors of the interpolation space.

Contribution

It provides a comparison between Sobolev--Slobodecki
2f and interpolation-based fractional Sobolev spaces, identifying when they are equivalent and highlighting unusual properties.

Findings

The two fractional Sobolev spaces do not always coincide.

Sufficient conditions are provided for their equivalence.

The interpolation space exhibits some unnatural behaviors.

Abstract

We consider a homogeneous fractional Sobolev space obtained by completion of the space of smooth test functions, with respect to a Sobolev--Slobodecki\u{\i} norm. We compare it to the fractional Sobolev space obtained by the $K-$method in real interpolation theory. We show that the two spaces do not always coincide and give some sufficient conditions on the open sets for this to happen. We also highlight some unnatural behaviors of the interpolation space. The treatment is as self-contained as possible.