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Logarithmic Sobolev inequalities on noncompact Riemannian manifolds. (English) Zbl 0887.35012

Summary: 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 (LSI) on noncompact manifolds. As a result, it is shown that LSI may hold even though the curvature of the operator is negative everywhere.

MSC:

35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
26D10 Inequalities involving derivatives and differential and integral operators
60J60 Diffusion processes
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