Probability Theory and Related Fields

, Volume 109, Issue 3, pp 417-424

First online:

Logarithmic Sobolev inequalities on noncompact Riemannian manifolds

  • Feng-Yu WangAffiliated withDepartment of Mathematics, Beijing Normal University, Beijing 100875, P. R. China e-mail:

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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.

AMS Subject Classification (1991): 35P15 60J60