Optimal Sobolev Inequalities of Arbitrary Order on Compact Riemannian Manifolds

Let (M, g) be a smooth compact RiemannianN-manifold,N⩾2, letp∈(1, N) real, and letHp1(M) be the Sobolev space of orderpinvolving first derivatives of the functions. By the Sobolev embedding theorem,Hp1(M)⊂Lp*(M) wherep*=Np/(N−p). Classically, this leads to some Sobolev inequality (I1p), and then to some Sobolev inequality (Ipp) where each term in (I1p) is elevated to the powerp. Long-standing questions were to know if the optimal versions with respect to the first constant of (I1p) and (Ipp) do hold. Such questions received an affirmative answer by Hebey–Vaugon forp=2. We prove here that forp>2, andp2<N, the optimal version of (Ipp) is false if the scalar curvature ofgis positive somewhere. In particular, there exist manifolds for which the optimal versions of (I1p) are true, while the optimal versions of (Ipp) are false. Among other results, we prove also that the assumption on the sign of the scalar curvature is sharp by showing that for anyp∈(1, N), the optimal version of (Ipp) holds on flat tori.