Log-Sobolev inequalities: Different roles of Ric and Hess

Abstract

Let $P_t$ be the diffusion semigroup generated by $L:=\Delta +\nabla V$ on a complete connected Riemannian manifold with $\mathrm{Ric}\ge -({\sigma }^2{\rho }_o^2+c)$ for some constants $\sigma ,c>0$ and ${\rho }_o$ the Riemannian distance to a fixed point. It is shown that $P_t$ is hypercontractive, or the log-Sobolev inequality holds for the associated Dirichlet form, provided $-{\mathrm{Hess}}_V\ge \delta$ holds outside of a compact set for some constant $\delta >(1+\sqrt{2})\sigma \sqrt{d-1}.$ This indicates, at least in finite dimensions, that $\mathrm{Ric}$ and $-{\mathrm{Hess}}_V$ play quite different roles for the log-Sobolev inequality to hold. The supercontractivity and the ultracontractivity are also studied.

Published in at http://dx.doi.org/10.1214/08-AOP444 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

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