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The Quarterly Journal of Mathematics Advance Access published online on October 26, 2006
The Quarterly Journal of Mathematics, doi:10.1093/qmath/hal021
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© 2006. Published by Oxford University Press. All rights reserved
Received May 17, 2006

Article

Optimal Integrability Condition for the log-Sobolev inequality

Xin Chen 1 and Feng-Yu Wang 1 *

1 School of Mathematical Sciences, Beijing Normal University, Beijing 100875, People's Republic of China

* To whom correspondence should be addressed.
Feng-Yu Wang, E-mail: wangfy{at}bnu.edu.cn


  Abstract

Let M denote a connected complete Riemannian manifold (possibly with a convex boundary), {rho} the Riemannian distance function from a fixed point and V  C2(M) such that dµV colone eV dx is a probability measure. For any K ≥ 0, we prove that K/2 is the infimum over all {lambda} > 0 such that RicM - HessV ≥ -K and µV(e{lambda}{rho}2) < {infty} imply the log-Sobolev inequality for the Dirichlet form µV(|{nabla}f|2).


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