  By using the mirror coupling for solutions of SDEs driven by pure jump L\'evy
processes, we extend some transportation and concentration inequalities, which
were previously known only in the case where the coefficients in the equation
satisfy a global dissipativity condition. Furthermore, by using the mirror
coupling for the jump part and the coupling by reflection for the Brownian
part, we extend analogous results for jump diffusions. To this end, we improve
some previous results concerning such couplings and show how to combine the
jump and the Brownian case. As a crucial step in our proof, we develop a novel
method of bounding Malliavin derivatives of solutions of SDEs with both jump
and Gaussian noise, which involves the coupling technique and which might be of
independent interest. The bounds we obtain are new even in the case of
diffusions without jumps.
