Logarithmic potentials on ${\mathbb{P}}^n$

We study the projective logarithmic potential ${\mathbb{G}}_{\mu }$ of a Probability measure $\mu$ on the complex projective space ${\mathbb{P}}^n$. We prove that the Range of the operator $\mu ⟶{\mathbb{G}}_{\mu }$ is contained in the (local) domain of definition of the complex Monge-Amp`ere operator acting on the class of quasi-plurisubharmonic functions on ${\mathbb{P}}^n$ with respect to the Fubini-Study metric. Moreover, when the measure $\mu$ has no atom, we show that the complex Monge-Amp`ere measure of its Logarithmic potential is an absolutely continuous measure with respect to the Fubini-Study volume form on ${\mathbb{P}}^n$