Logarithmic potentials on Pn

We study the projective logarithmic potential ${\mathbb{G}}_{\mu }$ of a probability measure μ 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–Ampère operator acting on the class of quasi-plurisubharmonic functions on ${\mathbb{P}}^n$ with respect to the Fubini–Study metric. Moreover, when the measure μ has no atom, we show that the complex Monge–Ampère measure of its logarithmic potential is an absolutely continuous measure with respect to the Fubini–Study volume form on ${\mathbb{P}}^n$.