Article

A family of functional inequalities: Lojasiewicz inequalities and displacement convex functions

Adrien Blanchet, and Jérôme Bolte

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 Łojasiewicz inequalities. We also discuss the more general case of λ-convex functions and we provide a general convergence theorem for the corresponding gradient dynamics. Specialising our results to the Boltzmann entropy, we recover Otto-Villani's theorem asserting the equivalence between logarithmic Sobolev and Talagrand's inequalities. The choice of power-type entropies shows a new equivalence between Gagliardo-Nirenberg inequality and a nonlinear Talagrand inequality. Some nonconvex results and other types of equivalences are discussed.

Keywords

Lojasiewicz inequality; Functional inequalities; Gradient flows; Optimal Transport; Monge-Kantorovich distance;

Replaces

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

Reference

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.

Published in

Journal of Functional Analysis, vol. 25, n. 7, October 2018, pp. 1650–1673