Equivalence of optimal L1-inequalities on Riemannian manifolds

Let (M,g) be a smooth compact Riemannian manifold of dimension n≥2. This paper concerns the validity of the optimal Riemannian L1-Entropy inequalityEntdvg(u)≤nlog⁡(Aopt‖Du‖BV(M)+Bopt) for all u∈BV(M) with ‖u‖L1(M)=1 and existence of extremal functions. In particular, we prove that this optimal inequality is equivalent to an optimal L1-Sobolev inequality obtained by Druet [3].