  Let $L$ be a second order elliptic operator on $R^d$ with a constant
diffusion matrix and a dissipative (in a weak sense) drift $b \in L^p_{loc}$
with some $p>d$.
  We assume that $L$ possesses a Lyapunov function, but no local boundedness of
$b$ is assumed. It is known that then there exists a unique probability measure
$\mu$ satisfying the equation $L^*\mu=0$ and that the closure of $L$ in
$L^1(\mu)$ generates a Markov semigroup $\{T_t\}_{t\ge 0}$ with the resolvent
$\{G_\lambda\}_{\lambda > 0}$.
  We prove that, for any Lipschitzian function $f\in L^1(\mu)$ and all
$t,\lambda>0$, the functions $T_tf$ and $G_\lambda f$ are Lipschitzian and
|\nabla T_tf(x)| \leq T_t|\nabla f|(x) and |\nabla G_\lambda f(x)| \leq
\frac{1}{\lambda} G_\lambda |\nabla f|(x).
  An analogous result is proved in the parabolic case.
