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.

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