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## $L^1$-convergence and hypercontractivity of diffusion semigroups on manifolds

### Tom 162 / 2004

Studia Mathematica 162 (2004), 219-227 MSC: 47D07, 60J60. DOI: 10.4064/sm162-3-3

#### Streszczenie

Let $P_t$ be the Markov semigroup generated by a weighted Laplace operator on a Riemannian manifold, with $\mu$ an invariant probability measure. If the curvature associated with the generator is bounded below, then the exponential convergence of $P_t$ in $L^1(\mu )$ implies its hypercontractivity. Consequently, under this curvature condition $L^1$-convergence is a property stronger than hypercontractivity but weaker than ultracontractivity. Two examples are presented to show that in general, however, $L^1$-convergence and hypercontractivity are incomparable.

#### Autorzy

• Feng-Yu WangDepartment of Mathematics
Beijing Normal University
Beijing 100875, P.R. China
e-mail

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