LOG-SOBOLEV INEQUALITIES: DIFFERENT ROLES OF RIC AND HESS

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