Sobolev–Hardy space with general weight

In this paper, it is defined the kth order Sobolev–Hardy space H1 0,k(Ω,φ) with norm u 1,k,φ = Ω φ|∇u|2 −φ k i=1 h i hi 2 u2 dx 1/2 . Then the corresponding Poincaré inequality in this space is obtained, and the results are given that this space is embedded in L 2N N−2 with weight φ−1|x|−2(N−1)H−(2+ 2N N−2 ) k+1 and in W 1,q 0 with weight φq/2 for 1 q <2. Moreover, we prove that the constant of k-improved Hardy–Sobolev inequality with general weight is optimal. These inequalities turn to be some known versions of Hardy–Sobolev inequalities in the literature by some particular choice of weights. © 2005 Elsevier Inc. All rights reserved