For \(p \in (1,N)\) and \(\Omega \subseteq {\mathbb {R}}^N\) open, the Beppo-Levi space \({\mathcal {D}}^{1,p}_0(\Omega )\) is the completion of \(C_c^{\infty }(\Omega )\) with respect to the norm \(\left[ \int _{\Omega }|\nabla u|^p \ dx \right] ^ \frac{1}{p}.\) Using the p-capacity, we define a norm and then identify the Banach function space \({\mathcal {H}}(\Omega )\) with the set of all g in \(L^1_{loc}(\Omega )\) that admits the following Hardy–Sobolev type inequality:

for some \(C>0.\) Further, we characterize the set of all g in \({\mathcal {H}}(\Omega )\) for which the map \(G(u)= \displaystyle \int _{\Omega } g |u|^p \ dx\) is compact on \({\mathcal {D}}^{1,p}_0(\Omega )\). We use a variation of the concentration compactness lemma to give a sufficient condition on \(g\in {\mathcal {H}}(\Omega )\) so that the best constant in the above inequality is attained in \({\mathcal {D}}^{1,p}_0(\Omega )\).

对于\（p \ in（1，N）\）和\（\ Omega \ subseteq {\ mathbb {R}} ^ N \）打开，Beppo-Levi空间\（{\ mathcal {D}} ^ {1 ，p} _0（\ Omega）\）是\（C_c ^ {\ infty}（\ Omega）\）相对于规范\（\ left [\ int _ {\ Omega} | \ nabla u | ^ p \ dx \ right] ^ \ frac {1} {p}。\）使用p -capacity定义一个范数，然后标识Banach函数空间\（{\ mathcal {H}}（\ Omega} \ ）与该组所有的克在\（L ^ 1_ {LOC}（\欧米茄）\） ，其接纳以下的Hardy-Sobolev型不等式：

对于一些\（C> 0 \）此外，我们表征集合中的所有的克在\（{\ mathcal {H}}（\欧米茄）\）的量，图\（G（U）= \的DisplayStyle \ INT _ {\ Omega} g | u | ^ p \ dx \）在\（{\ mathcal {D}} ^ {1，p} _0（\ Omega）\）上是紧凑的。我们使用浓度紧凑性引理的一个变体来给\（g \ in {\ mathcal {H}}（\ Omega）\）给出足够的条件，从而在\（{\ mathcal {D}} ^ {1，p} _0（\ Omega）\）。