Extremal omega-plurisubharmonic functions as envelopes of disc functionals - Generalization and applications to the local theory

We generalize the Poletsky disc envelope formula for the function $\sup \{u \in \PSH(X,\omega) ; u\leq \phi\}$ on any complex manifold $X$ to the case where the real (1,1)-current $\omega=\omega_1-\omega_2$ is the difference of two positive closed (1,1)-currents and $\phi$ is the difference of an $\omega_1$-upper semicontinuous function and a plurisubharmonic function.