*Accepted Paper*

**Inserted:** 15 jul 2020

**Last Updated:** 31 may 2021

**Journal:** Calc. Var. Partial Differential Equations

**Pages:** 31

**Year:** 2020

**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.

**Keywords:**
fractional Sobolev spaces, Gagliardo-Slobodecki\u{\i} norms, embeddings, H\"older spaces, BMO space, Campanato spaces

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