Journal
POTENTIAL ANALYSIS
Volume 36, Issue 2, Pages 317-337Publisher
SPRINGER
DOI: 10.1007/s11118-011-9232-2
Keywords
Logarithmic-Sobolev inequalites; Talagrand inequalites; Hamilton-Jacobi semigroup; Poincare inequalities; Geodesic metric space; Metric-measure space
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Funding
- Swiss Nationalfond
- EC
- ERC
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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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