Weak Dirichlet processes with jumps

This paper develops systematically the stochastic calculus via regularization in the case of jump processes. In particular one continues the analysis of real-valued càdlàg weak Dirichlet processes with respect to a given filtration. Such a process is the sum of a local martingale and an adapted process A such that [N,A]=0, for any continuous local martingale N. Given a function u:[0,T]×R→R, which is of class C^0,1 (or sometimes less), we provide a chain rule type expansion for u(t,X_t) which stands in applications for a chain Itô type rule.