A note on Poincar\'e-Sobolev type inequalities on compact manifolds

We prove a Poincar\'e-Sobolev type inequality on compact Riemannian manifolds where the deviation of a function from a biased average, defined using a density, is controlled by the unweighted Lebesgue norm of its gradient. Unlike classical weighted Poincar\'e inequalities, the density does not enter the measure or the Sobolev norms, but only the reference average. We show that the associated Poincar\'e constant depends quantitatively on the Lebesgue norm of the density. This framework naturally arises in the analysis of coupled elliptic systems and seems not to have been addressed in the existing literature.