Characterisation of homogeneous fractional Sobolev spaces

Our aim is to characterize the homogeneous fractional Sobolev–Slobodeckiĭ 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–Slobodeckiĭ 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–Slobodeckiĭ seminorm controls from above a suitable Campanato seminorm.