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.
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