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

## TL;DR

This paper establishes a link between Sobolev inequalities and exponential convergence of Markov diffusion semigroups in -entropy, providing estimates and perturbation results, with specific insights in the one-dimensional case.

## Contribution

It demonstrates the equivalence between Sobolev inequalities and exponential convergence in -entropy, offering new estimates and perturbation results, especially in one dimension.

## Key findings

- Sobolev inequality is equivalent to exponential convergence in -entropy.
- Provides estimates of convergence in total variation.
- Derives two-sided bounds of the Sobolev constant via Hardy inequality in 1D.

## 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.