Abstract
In this short paper we find that the Sobolev inequality 1p−2∫fpdμ2p−∫f2dμ≤C∫|∇f|2dμ(p≥0) is equivalent to the exponential convergence of the Markov diffusion semigroup (Pt) 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.
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