  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.
