We prove that in the context of general Markov semigroups Beckner inequalities with constants separated from zero as p → 1 + are equivalent to the modified log Sobolev inequality (previously only one implication was known to hold in this generality). Further, by adapting an argument by Boucheron et al. we derive Sobolev type moment estimates which hold under these functional inequalities.We illustrate our results with applications to concentration of measure estimates (also of higher order, beyond the case of Lipschitz functions) for various stochastic models, including random permutations, zero-range processes, strong Rayleigh measures, exponential random graphs, and geometric functionals on the Poisson path space.
Contents1. Introduction 1.1. Motivation and informal presentation 1.2. General setting 1.3. Examples 1.4. Functional inequalities 2. From modified log-Sobolev to Beckner's inequalities 2.1. Main result 2.2. Auxiliary lemmas 2.3. Proof of Theorem 2.1 3. Moment estimates derived from Beckner's inequalities 4. Applications 4.1. The continuous setting 4.2. Product spaces 4.3. Glauber dynamics 4.4. The symmetric group 4.5. Stochastic covering property 4.6. Zero-range processes 4.7. The Poisson space 5. Higher order concentration 5.1. Abstract inequality 5.2. Discussion on the choice of gradients 5.3. Applications to tetrahedral polynomials Appendix A.