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This paper deals with fractional Sobolev spaces on a compact Riemannianmanifold. We prove a Sobolev inequality in the critical range with an optimalconstant for these fractional Sobolev spaces. We use this result to study theexistence of a non-trivial solution for equations driven by a non-localintegro-differential operator $\mathcal{L}_{\mathcal{K}}$ with criticalnon-linearity.

Non-local equations and optimal Sobolev inequalities on compact manifolds