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

Feng-Yu Wang¹

¹ Department of Mathematics Beijing Normal University Beijing 100875, P.R. China

Studia Mathematica, Tome 162 (2004) no. 3, pp. 219-227

Résumé

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.

Détail

DOI : 10.4064/sm162-3-3

Keywords: markov semigroup generated weighted laplace operator riemannian manifold invariant probability measure curvature associated generator bounded below exponential convergence implies its hypercontractivity consequently under curvature condition convergence property stronger hypercontractivity weaker ultracontractivity examples presented general however convergence hypercontractivity incomparable