Abstract
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 pge 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.
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