Optimal Hardy–Sobolev inequalities on compact Riemannian manifolds

Given a compact Riemannian manifold (M,g)(M,g) of dimension n≥3n≥3, a point x0∈Mx0∈M and s∈(0,2)s∈(0,2), the Hardy–Sobolev embedding yields the existence of A,B>0A,B>0 such thatequation(1)(∫M|u|2(n−s)n−2dg(x,x0)sdvg)n−2n−s≤A∫M|∇u|g2dvg+B∫Mu2dvg for all u∈H12(M). It has been proved in Jaber [20] that A≥K(n,s)A≥K(n,s) and that one can take any value A>K(n,s)A>K(n,s) in (1), where K(n,s)K(n,s) is the best possible constant in the Euclidean Hardy–Sobolev inequality. In the present manuscript, we prove that one can take A=K(n,s)A=K(n,s) in (1).