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 c`adl`ag 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 $u:[0,T]\times \mathbb{R}\to \mathbb{R}$, 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 It^o type rule.