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 refinement of the Bakry-A parts per thousand mery method is established, which allows us to prove non-trivial inequalities even in situations where the classical Bakry-A parts per thousand mery criterion fails. The main application of our theory concerns the linearized fast diffusion equation in dimensions d a parts per thousand 1, which admits a Poincar,, 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 specified range. In dimension d = 1, our estimates improve the corresponding results that can be obtained by the measure-theoretic techniques of Barthe and Roberto. As a by-product, we give a short and elementary alternative proof of the sharp spectral gap inequality first obtained by Denzler and McCann. In further applications of our method, we prove convex Sobolev inequalities for a mean field model for the redistribution of wealth in a simple market economy, and the Lasota model for blood cell production.