A family of functional inequalities

Abstract

For displacement convex functionals in the probability space equipped with the Monge-Kantorovich metric we prove the equivalence between the gradient and functional type Lojasiewicz inequalities. In a second part, we specialise these inequalities to some classical geodesically convex functionals. For the Boltzmann entropy, we obtain the equivalence between logarithmic Sobolev and Talagrand's inequalities. On the other hand, the non-linear entropy and the Gagliardo-Nirenberg inequality provide a Talagrand inequality which seems to be a new equivalence. Our method allows also to recover some results on the asymptotic behaviour of the associated gradient flows.

Replaced by

Adrien Blanchet, and Jérôme Bolte, “A family of functional inequalities: Lojasiewicz inequalities and displacement convex functions”, Journal of Functional Analysis, vol. 25, n. 7, October 2018, pp. 1650–1673.

Reference

Adrien Blanchet, and Jérôme Bolte, “A family of functional inequalities: lojasiewicz inequalities and displacement convex functions”, TSE Working Paper, n. 17-787, March 2017.