Smoothness of harmonic functions for processes with jumps

We consider a non-local operator L associated to a Markov process with jumps, we stop this process when it quits a domain D, and we study the Cj smoothness on D of the functions which are harmonic for the stopped process. A previous work was devoted to the existence of a C∞ transition density; here, the smoothness of harmonic functions is deduced by applying a duality method and by estimating the density in small time.