Centered Sobolev inequality and exponential convergence in $\Phi$-entropy

Abstract

In this short paper we find that the Sobolev inequality $$\frac{1}{p-2}\left[{\left(\int f^{p}d\mu \right)}^{2/p}-\int f^{2}d\mu \right]\le C\int |\nabla f{|}^{2}d\mu$$ ($p\ge 0$) is equivalent to the exponential convergence of the Markov diffusion semigroup $(P_t)$ to the invariant measure $\mu$, in some $\Phi$-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.

14 pages