OPTIMAL INTEGRABILITY CONDITION FOR THE LOG-SOBOLEV INEQUALITY

Let M denote a connected complete Riemannian manifold (possibly with a convex boundary), ρ the Riemannian distance function from a fixed point and V ∈ C² (M) such that dμV≔ e^V d x is a probability measure. For any K ≥ 0, we prove that K/2 is the infimum over all λ > 0 such that RicM − HessV ≥ −K and $\mu \rm v({\rm e^{\lambda\rho}}^2)$ imply the log-Sobolev inequality for the Dirichlet form μV(|∇ f |²).