Characterisation of homogeneous fractional Sobolev spaces

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 $sp<n$ or $s=p=n=1$ we show that ${\mathcal{D}}^{s,p}({\mathbb{R}}^n)$ is isomorphic to a suitable function space, whereas for $sp\ge n$ 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.