Projective Logarithmic Potentials

We study the projective logarithmic potential $G_\mu$ of a Probability measure $\mu$ on the complex projective space ${P}^{n}$ equiped with the Fubini-Study metric $\omega$. We prove that the Green operator $G$ has strong regularizing properties. It was shown by the second author that the range of the operator $G$ is contained in the (local) domain of definition of the complex Monge-Amp\`ere operator on $P^n$. This result extends earlier results by Carlehed. We will show that the complex Monge-Amp\`ere measure of the logarithmic potential of $\mu$ is absolutely continuous with respect to the Lebesgue measure on $P^n$ if and only if the measure $\mu$ has no atoms. Moreover when the measure $\mu$ has a "positive dimension", we give more precise results on regularity properties of the potential $G_\mu$ in terms of the dimension of $\mu$.