Growth description of $p$th means of the Green potential in the unit ball

We describe the growth of $p$th means, $1<p<\frac{2n-1}{2(n-1)}$, of the invariant Green potential in the unit ball in ${\mathbb{C}}^n$ in terms of smoothness properties of a measure. In particular, a criterion of boundedness of $p$th means of the potential is obtained, a result of M. Stoll is generalized.