Analytic properties of Markov semigroup generated by Stochastic Differential Equations driven by L\'evy processes

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 $({\mathcal{P}}_t{)}_{t\ge 0}$ 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 $$\mid B{\mathcal{P}}_{t}u{\mid }_{H_2^{\rho }}\le C{t}^{-\gamma }\mid u{\mid }_{H_2^{\rho }},\mathrm{\forall }u\in H_2^{\rho }({\mathbb{R}}^d),t>0.$$