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-infinity(M) given by L(U)f = Delta f-(del U\del f), wh ere U is an element of C-infinity(M), M being a finite dimensional compact Riemannian manifold without boundary, in terms of the spectral gap of LU an d the lowest eigenvalue of the operator -L-U + V, where V is a function rel ated to U and the Ricci curvature of M. Under suitable conditions and being U equivalent to 0, this result improves a previous one by J.-D. DEUSCHEL a nd D.W. STROOCK (J. Funct. Anal. 92 (1990), 30-48).