Lower bounds for the logarithmic Sobolev constant avoiding uniform lower bounds on the Ricci curvature

In this paper we obtain a lower bound for the logarithmic Sobolev constant of the operator on C^∞(M) given by L^U f = Δ f - (?U|?f), where U ? C^∞(M), M being a finite dimensional compact Riemannian manifold without boundary, in terms of the spectral gap of L^U and the lowest eigenvalue of the operator -L^U + V, where V is a function related to U and the Ricci curvature of M. Under suitable conditions and being U ≡ 0, this result improves a previous one by J.-D. DEUSCHEL and D.W. STROOCK (J. Funct. Anal. 92 (1990), 30–48).