Note on the q-logarithmic Sobolev and p-Talagrand inequalities on Carnot groups

Abstract

In the setting of Carnot groups, we prove the [Formula: see text]-logarithmic Sobolev inequality for probability measures as a function of the Carnot–Carathéodory distance. As an application, we use the Hamilton–Jacobi equation in the setting of Carnot groups to prove the [Formula: see text]-Talagrand inequality and hypercontractivity.