  For Anderson Localization models with multifractal eigenvectors on disordered
samples containing $N$ sites, we analyze in a unified framework the
consequences for the statistical properties of the Green function. We focus in
particular on the imaginary part of the Green function at coinciding points
$G^I_{xx}(E-i \eta)$ and study the scaling with the size $N$ of the moments of
arbitrary indices $q$ when the broadening follows the scaling
$\eta=\frac{c}{N^{\delta}}$. For the standard scaling regime $\delta=1$, we
find in the two limits $c \ll 1$ and $c \gg 1$ that the moments are governed by
the anomalous exponents $\Delta(q)$ of individual eigenfunctions, without the
assumption of strong correlations between the weights of consecutive
eigenstates at the same point. For the non-standard scaling regimes
$0<\delta<1$, we obtain that the imaginary Green function follows some
Fr\'echet distribution in the typical region, while rare events are important
to obtain the scaling of the moments. We describe the application to the case
of Gaussian multifractality and to the case of linear multifractality.
