Functional Inequalities and Hamilton-Jacobi Equations in Geodesic Spaces

We study the connection between the p-Talagrand inequality and the q-logarithmic Sololev inequality for conjugate exponents p >= 2, q <= 2 in proper geodesic metric spaces. By means of a general Hamilton-Jacobi semigroup we prove that these are equivalent, and moreover equivalent to the hypercontractivity of the Hamilton-Jacobi semigroup. Our results generalize those of Lott and Villani. They can be applied to deduce the p-Talagrand inequality in the sub-Riemannian setting of the Heisenberg group.