, Volume 109, Issue 3, pp 417-424

Logarithmic Sobolev inequalities on noncompact Riemannian manifolds

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This paper presents a dimension-free Harnack type inequality for heat semigroups on manifolds, from which a dimension-free lower bound is obtained for the logarithmic Sobolev constant on compact manifolds and a new criterion is proved for the logarithmic Sobolev inequalities (abbrev. LSI) on noncompact manifolds. As a result, it is shown that LSI may hold even though the curvature of the operator is negative everywhere.

Received: 24 July 1996 / In revised form: 25 June 1997