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Journal of Operator Theory

Volume 46, Issue 1, Summer 2001  pp. 183-197.

Logarithmic Sobolev inequalities: conditions and counterexamples

Authors:  Feng-Yu Wang
Author institution: Department of Mathematics, Beijing Normal University, Beijing 100875, P.R. China

Summary:  Let $M$ be any noncompact, connected, complete Riemannian manifold with Riemannian distance function (from a fixed point) $\rr$. Consider $L= \DD +\nn V$ for some $V\in C^2(M)$ with $\d \mu:= \e^V\d x$ a probability measure. Define $\delta\ge 0$ as the smallest possible constant such that for any $K$, $\vv>0$, $\mu(\exp[(\dd K +\vv)\rr^2])<\infty$ implies the logarithmic Sobolev inequality (abbrev. LSI) for any $M$ and $V$ with Ric-Hess$_V\ge -K.$ It is shown in the paper that $\dd \in [\ff 1 4, \ff 1 2].$ Moreover, some differential type conditions are presented for the LSI. As a consequence, a result suggested by D. Stroock is proved: for $V=-r \rr^2$ with $r>0$, the LSI holds provided the Ricci curvature is bounded below.

Keywords:  Logarithmic Sobolev inequality, Riemannian manifold, Ricci curvature

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