Role of the fundamental solution in Hardy-Sobolev-type inequalities

Abstract

Let n ≥ 3, Ω⊂ Rⁿ be a domain with 0∈ Ω, then, for all u∈ H¹₀ Ω, the Hardy-Sobolev inequality says that ∫_Ω|∇u|²-(n-2/2)² ∫_Ωu²/|x|²≥0 and equality holds if and only if u = 0 and ((n-2)/2)² is the best constant which is never achieved. In view of this, there is scope for improving this inequality further. In this paper we have investigated this problem by using the fundamental solutions and have obtained the optimal estimates. Furthermore, we have shown that this technique is used to obtain the Hardy-Sobolev type inequalities on manifolds and also on the Heisenberg group.