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The Quarterly Journal of Mathematics Advance Access originally published online on October 26, 2006
The Quarterly Journal of Mathematics 2007 58(1):17-22; doi:10.1093/qmath/hal021
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© 2006. Published by Oxford University Press. All rights reserved. For permissions, please email: journals.permissions@oxfordjournals.org

OPTIMAL INTEGRABILITY CONDITION FOR THE LOG-SOBOLEV INEQUALITY

Xin Chen and Feng-Yu Wang{dagger}

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

{dagger} Corresponding author. E-mail: wangfy{at}bnu.edu.cn

Received 17 May 2006;
  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 isin C2 (M) such that dµVcolone eV d x 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 Formula imply the log-Sobolev inequality for the Dirichlet form µV(|{nabla} f |2).


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