  We consider the stochastic differential equations of the form
\begin{equation*} \begin{cases} dX^ x(t) = \sigma(X(t-)) dL(t) \\ X^
x(0)=x,\quad x\in\mathbb{R}^ d, \end{cases} \end{equation*} where
$\sigma:\mathbb{R}^ d\to \mathbb{R}^ d$ is Lipschitz continuous and
$L=\{L(t):t\ge 0\}$ is a L\'evy process. Under this condition on $\sigma$ it is
well known that the above problem has a unique solution $X$. Let
$(\mathcal{P}_{t})_{t\ge0}$ be the Markovian semigroup associated to $X$
defined by $( \mathcal{P}_t f) (x) := \mathbb{E} [ f(X^ x(t))]$, $t\ge 0$,
$x\in \mathbb{R}^d$, $f\in \mathcal{B}_b(\mathbb{R}^d)$. Let $B$ be a
pseudo--differential operator characterized by its symbol $q$. Fix
$\rho\in\mathbb{R}$. In this article we investigate under which conditions on
$\sigma$, $L$ and $q$ there exist two constants $\gamma>0$ and $C>0$ such that
$$ \lvert B \mathcal{P}_t u \rvert_{H^\rho_2} \le C \, t^{-\gamma} \,\lvert u
\rvert_{H^\rho_2}, \quad \forall u \in {H^\rho_2}(\mathbb{R}^d ),\, t>0. $$
