L¹-convergence and hypercontractivity of diffusion semigroups on manifolds

Feng-Yu Wang. "L¹-convergence and hypercontractivity of diffusion semigroups on manifolds." Studia Mathematica 162.3 (2004): 219-227. <http://eudml.org/doc/284450>.

@article{Feng2004,
abstract = {Let $P_\{t\}$ 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 $P_\{t\}$ 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.},
author = {Feng-Yu Wang},
journal = {Studia Mathematica},
keywords = {semigroups; diffusion processes; hypercontractivity; ultracontractivity; exponential convergence; Riemannian manifolds},
language = {eng},
number = {3},
pages = {219-227},
title = {L¹-convergence and hypercontractivity of diffusion semigroups on manifolds},
url = {http://eudml.org/doc/284450},
volume = {162},
year = {2004},
}

TY - JOUR
AU - Feng-Yu Wang
TI - L¹-convergence and hypercontractivity of diffusion semigroups on manifolds
JO - Studia Mathematica
PY - 2004
VL - 162
IS - 3
SP - 219
EP - 227
AB - Let $P_{t}$ 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 $P_{t}$ 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.
LA - eng
KW - semigroups; diffusion processes; hypercontractivity; ultracontractivity; exponential convergence; Riemannian manifolds
UR - http://eudml.org/doc/284450
ER -