Sobolev–Hardy space with general weight

In this paper, it is defined the k  th order Sobolev–Hardy space H0,k1(Ω,ϕ) with norm‖u‖1,k,ϕ={∫Ω[ϕ|∇u|2−ϕ∑i=1k(hi′hi)2u2]dx}1/2. Then the corresponding Poincaré inequality in this space is obtained, and the results are given that this space is embedded in L2NN−2 with weight ϕ−1|x|−2(N−1)Hk+1−(2+2NN−2) and in W01,q with weight ϕq/2ϕq/2 for 1⩽q<21⩽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.