Remarks on logarithmic Sobolev constants, exponential integrability and bounds on the diameter

We present short and elementary proofs of three recent results on exponential integrability of Lipschitz functions and quantitative bounds on the diameter under logarithmic Sobolev inequalities due respectively to S. Aida, T. Masuda, I. Shigekawa [A-M-S], D. Bakry, D. Michel [B-M] and L. Saloff-Coste [SC]. Although the first two results we aim to simplify deal with abstract Markov generators on probability spaces, we would like to briefly present the purpose of this note in the setting of the Laplace-Beltrami operator ∆ on a complete connected Riemannian manifold M of finite volume V . We will consider the normalized measure dμ = 1 V dv where dv denote the Riemannian measure and let ∇ be the Riemannian gradient on M . For a nonnegative bounded (say) real valued function f on M , let E(f) denote the entropy of f with respect to μ defined by E(f) = ∫ f log fdμ− ∫ fdμ log (∫ fdμ ) . We will say that ∆ satisfies a logarithmic Sobolev inequality if there exists ρ > 0 such that for all C∞, compactly supported or bounded, functions f on M , ρE(f) ≤ 2 ∫ f(−∆f)dμ = 2 ∫ |∇f |dμ. The largest possible value ρ0 for ρ is called the logarithmic Sobolev constant of the Laplacian ∆ on M , or simply of M . More generaly, one may consider, following [B], inequalities between entropy and energy of the type E(f) ≤ Φ (∥∥|∇f |∥∥2 2 ) for all C∞ bounded functions f with ‖f‖2 = 1 where Φ is a nonnegative function on [0,∞). With these notations, S. Aida, T. Masuda and I. Shigekawa [A-M-S] recently showed that, when ρ0 > 0, whenever f is a function on M such that ‖|∇f |‖∞ ≤ 1 (that is f is Lipschitz with Lipschitz norm less than or equal to 1), then, for every 0 < α < ρ0/2