Characterisation of homogeneous fractional Sobolev spaces

TL;DR

This paper characterizes homogeneous fractional Sobolev spaces and their embeddings, revealing their structure as function spaces or classes of functions differing by constants depending on parameters, using a Morrey-Campanato inequality.

Contribution

It provides a detailed characterization of homogeneous fractional Sobolev spaces and their embeddings, including new isomorphism results based on parameter regimes.

Findings

For s p < n, the space is isomorphic to a function space.

For s = p = n = 1, the space is isomorphic to a space of equivalence classes.

A Morrey-Campanato inequality is established linking seminorms.

Abstract

Our aim is to characterize the homogeneous fractional Sobolev-Slobodecki\u{\i} spaces $\mathcal{D}^{s,p} (\mathbb{R}^n)$ and their embeddings, for $s \in (0,1]$ and $p\ge 1$. They are defined as the completion of the set of smooth and compactly supported test functions with respect to the Gagliardo-Slobodecki\u{\i} seminorms. For $s\,p < n$ or $s = p = n = 1$ we show that $\mathcal{D}^{s,p}(\mathbb{R}^n)$ is isomorphic to a suitable function space, whereas for $s\,p \ge n$ it is isomorphic to a space of equivalence classes of functions, differing by an additive constant. As one of our main tools, we present a Morrey-Campanato inequality where the Gagliardo-Slobodecki\u{\i} seminorm controls from above a suitable Campanato seminorm.