Functional inequalities and Hamilton-Jacobi Equations in Geodesic Spaces

2009/06/02 by Balogh, Zoltan, Engoulatov, Alexandre, Hunziker, Lars +1
#47D06 #49L99 #70H20 #FOS: Mathematics #Functional Analysis (math.FA) #Metric Geometry (math.MG)

paper · doi:10.48550/arxiv.0906.0476

Abstract

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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