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 \(p\ge 1\). 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.

我们的目的是表征均匀分数的Sobolev-Slobodeckiĭ空间\（\ mathcal {d} ^ {S，P}（\ mathbb {R} ^ N）\）和它们的嵌入物，为\（S \在（0,1 ] \）和\（p \ GE 1 \） ，它们被定义为一组的光滑和紧支撑测试功能的完成相对于所述Gagliardo-Slobodeckiĭ半范。对于\（S \，p <N \）或\（s = p = n = 1 \）我们证明\（\ mathcal {D} ^ {s，p}（\ mathbb {R} ^ n）\）同构到合适的函数空间，而对于\（ s \，p \ ge n \）它对等价的函数类空间是同构的，只是相加常数不同。作为我们的主要工具之一，我们提出了一个Morrey-Campanato不等式，其中Gagliardo-Slobodeckiĭ半范数是从合适的Campanato半范数上控制的。