Gradient formula for transition semigroup corresponding to stochastic equation driven by a system of independent L\'evy processes

Let $(P_t)$ be the transition semigroup of the Markov family $(X^x(t))$ defined by SDE $$dX=b(X)dt+dZ,X(0)=x,$$ where $Z=\left(Z_1, \ldots, Z_d\right)^*$ is a system of independent real-valued L'evy processes. Using the Malliavin calculus we establish the following gradient formula $$\mathrm{\nabla }{P}_{t}f(x)=\mathbb{E}f(X^x(t))Y(t,x),f\in B_b({\mathbb{R}}^d),$$ where the random field $Y$ does not depend on $f$. Sharp estimates on $\mathrm{\nabla }{P}_{t}f(x)$ when $Z_1,\dots ,Z_d$ are $\alpha$-stable processes, $\alpha \in (0,2)$, are also given.