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

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