Characterisation of homogeneous fractional Sobolev spaces

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