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 $p\geq 2$, $q\leq 2$ 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.