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