Hardy–Littlewood–Sobolev inequalities on compact Riemannian manifolds and applications

In this paper we extend Hardy–Littlewood–Sobolev inequalities on compact Riemannian manifolds for dimension n≠2n≠2. As one application, we solve a generalized Yamabe problem on locally conformally flat manifolds via a new designed energy functional and a new variational approach. Even for the classic Yamabe problem on locally conformally flat manifolds, our approach provides a new and relatively simpler solution.