Gradient formula for transition semigroup corresponding to stochastic equation driven by a system of independent Levy processes

Let (P-t) be the transition semigroup of the Markov family (X-x (t)) defined by SDEdX = b(X)dt+ dZ, X(0) = x,where Z = (Z(1),..., Z(d))* is a system of independent real-valued Levy processes. Using the Malliavin calculus we establish the following gradient formuladel P(t)f(x) = E f (X-x (t)) Y (t, x), f is an element of B-b(R-d),where the random field Y does not depend on f. Moreover, in the important cylindrical alpha-stable case alpha is an element of (0, 2), where Z(1),..., Z(d) are alpha-stable processes, we are able to prove sharp L-1-estimates for Y (t, x). Uniform estimates on del P(t)f(x) are also given.