A family of Beckner inequalities under various curvature-dimension conditions

In this paper, we offer a proof for a family of functional inequalities interpolating between the Poincar{\'e} and the logarithmic Sobolev (standard and weighted) inequalities. The proofs rely both on entropy flows and on a CD($\rho$, n) condition, either with $\rho$ = 0 and n > 0, or with $\rho$ > 0 and n $\in$ R. As such, results are valid in the case of a Riemannian manifold, which constitutes a generalization to what was proved in [BGS18, Ngu18].