Characterisation of homogeneous fractional Sobolev spaces

Abstract

AbstractOur aim is to characterize the homogeneous fractional Sobolev–Slobodeckiĭ spaces $$\mathcal {D}^{s,p} (\mathbb {R}^n)$$ D s , p ( R n ) and their embeddings, for $$s \in (0,1]$$ s ∈ ( 0 , 1 ] and $$p\ge 1$$ p ≥ 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 $$s\,p < n$$ s p < n or $$s = p = n = 1$$ s = p = n = 1 we show that $$\mathcal {D}^{s,p}(\mathbb {R}^n)$$ D s , p ( R n ) is isomorphic to a suitable function space, whereas for $$s\,p \ge n$$ s p ≥ 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.