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 $(\mathcal{P}_{t})_{t\ge0}$ 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 $$\lvert B \mathcal{P}_t u \rvert_{H^\rho_2} \le C \, t^{-\gamma} \,\lvert u \rvert_{H^\rho_2}, \quad \forall u \in {H^\rho_2}(\mathbb{R}^d ),\, t>0.$$