Abstract :
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
