  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.
