Optimal Hardy-Sobolev Inequalities on Compact Riemannain Manifolds

TL;DR

This paper proves the existence of optimal Hardy-Sobolev inequalities on compact Riemannian manifolds, establishing that the best Euclidean constant can be achieved in the manifold setting.

Contribution

It demonstrates that the optimal Hardy-Sobolev constant from Euclidean space extends to compact Riemannian manifolds, filling a gap in the understanding of these inequalities.

Findings

Optimal Hardy-Sobolev inequalities hold on compact Riemannian manifolds.

The best Euclidean constant is attainable on manifolds.

The inequality constants are sharp and optimal.

Abstract

Given a compact Riemannian Manifold (M,g) of dimension n > 2, a point x_0 in M and s in (0,2). We let 2*(s) = 2(n-s)/(n-2) be the critical Hardy-Sobolev exponent. The Hardy-Sobolev embedding yields the existence of A,B > 0 such that (\int_M|u|^{2*(s)}dv_g)^{2/2*(s)} \leq A\int_M |\nabla u|_g^2 dv_g +B\int_M u^2 dv_g for all u in H_1^2(M). It has been proved that A\leq K(n,s) and that one can take any value A > K(n,s) in in the above inequality where $K(n,s)$ is the best possible constant in the Euclidean Hardy-Sobolev inequality. In the present manuscript, we prove that one can also take A = K(n,s) in the above inequality.