Maximal subextensions of plurisubharmonic functions

In this paper we are concerned with the problem of local and global subextensions of (quasi-)plurisubharmonic functions from a “regular” subdomain of a compact K"ahler manifold. We prove that a precise bound on the complex Monge-Amp`ere mass of the given function implies the existence of a subextension to a bigger regular subdomain or to the whole compact manifold. In some cases we show that the maximal subextension has a well defined complex Monge-Amp`ere measure and obtain precise estimates on this measure. Finally we give an example of a plurisubharmonic function with a well defined Monge-Amp`ere measure and the right bound on its Monge-Amp`ere mass on the unit ball in ${\text{C}}^n$ for which the maximal subextension to the complex projective space $\text{mb}{P}_n$ does not have a globally well defined complex Monge-Amp`ere measure.