Centered Sobolev inequality and exponential convergence in [formula omitted]-entropy.

Abstract In this short paper we find that the Sobolev inequality 1 p − 2 ∫ f p d μ 2 p − ∫ f 2 d μ ≤ C ∫ | ∇ f | 2 d μ (p ≥ 0) is equivalent to the exponential convergence of the Markov diffusion semigroup (P t) to the invariant measure μ , in some Φ -entropy. We provide the estimate of the exponential convergence in total variation and a bounded perturbation result under the Sobolev inequality. Finally in the one-dimensional case we get some two-sided estimates of the Sobolev constant by means of the generalized Hardy inequality. [ABSTRACT FROM AUTHOR]