Logarithmic Sobolev inequalities: Conditions and counterexamples

Let M be any noncompact, connected, complete Riemannian manifold with Riema nnian distance function (from a fixed point) rho. Consider L = Delta + delV for some V is an element of C-2(M) with d mu := e(v) dx a probability meas ure. Define delta greater than or equal to 0 as the smallest possible const ant such that for any K, epsilon > 0, mu (exp[(deltaK + epsilon)rho (2)]) < <infinity> implies the logarithmic Sobolev inequality (abbrev. LSI) for an y M and V with Ric-Hess(v) greater than or equal to -K. It is shown in the paper that delta is an element of [1/4, 1/2]. Moreover, some differential type conditions are presented for the LSI. As a consequence, a result suggested by D. Stroock is proved: for V = -r rho (2 ) with r > 0, the LSI holds provided the Ricci curvature is bounded below.