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.
Categories: math.AP math.FA