Note on the $q$-Logarithmic Sobolev and $p$-Talagrand Inequalities on Carnot Groups

Abstract

In the setting of Carnot groups, we prove the $q-$Logarithmic Sobolev inequality for probability measures as a function of the Carnot-Carath\'eodory distance. As an application, we use the Hamilton-Jacobi equation in the setting of Carnot groups to prove the $p-$Talagrand inequality and hypercontractivity.