Let be the Markov semigroup generated by a weighted Laplace operator on a Riemannian manifold, with μ an invariant probability measure. If the curvature associated with the generator is bounded below, then the exponential convergence of in L¹(μ) implies its hypercontractivity. Consequently, under this curvature condition L¹-convergence is a property stronger than hypercontractivity but weaker than ultracontractivity. Two examples are presented to show that in general, however, L¹-convergence and hypercontractivity are incomparable.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-sm162-3-3,
author = {Feng-Yu Wang},
title = {L$^1$-convergence and hypercontractivity of diffusion semigroups on manifolds},
journal = {Studia Mathematica},
volume = {162},
year = {2004},
pages = {219-227},
zbl = {1084.58014},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm162-3-3}
}
Feng-Yu Wang. L¹-convergence and hypercontractivity of diffusion semigroups on manifolds. Studia Mathematica, Tome 162 (2004) pp. 219-227. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm162-3-3/