Log-Sobolev inequalities: Different roles of Ric and Hess

Abstract

Let P_t be the diffusion semigroup generated by L:=Δ+∇V on a complete connected Riemannian manifold with Ric≥−(σ²ρ_o²+c) for some constants σ, c>0 and ρ_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 −Hess_V≥δ 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 Ric and −Hess_V play quite different roles for the log-Sobolev inequality to hold. The supercontractivity and the ultracontractivity are also studied.