  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 $\mu$.
