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: = eV dx 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 µV (eλρ2) < ∞ imply the log-Sobolev inequality for the Dirichlet form µV (|∇f |2). 1