Convex Sobolev inequalities derived from entropy dissipation

We study families of convex Sobolev inequalities, which arise as entropy-dissipation relations for certain linear Fokker-Planck equations. Extending the ideas recently developed by the first two authors, a renement of the Bakry-Emery method is established, which allows us to prove non-trivial inequalities even in situations where the classical Bakry-Emery criterion fails. The main application of our theory concerns the linearized fast diusion equation in dimensions d = 1, which admits a Poincaré, but no logarithmic Sobolev inequality. We calculate bounds on the constants in the interpolating convex Sobolev inequalities, and prove that these bounds are sharp on a specied range. In dimension d = 1, our estimates improve the corresponding results that can be obtained by the measure-theoretic techniques of Barthe and Roberto. In further applications of our method, we prove convex Sobolev inequalities for a mean eld model for the redistribution of wealth in a simple market economy, and the Lasota model for blood cell production.The research that led to the present paper was partially supported by a grant of the group GNFM of INdAM.