Non-local equations and optimal Sobolev inequalities on compact manifolds

This paper deals with fractional Sobolev spaces on a compact Riemannian manifold. We prove a Sobolev inequality in the critical range with an optimal constant for these fractional Sobolev spaces. We use this result to study the existence of a non-trivial solution for equations driven by a non-local integro-differential operator ${\mathcal{L}}_{\mathcal{K}}$ with critical non-linearity.