Boundary behavior of invariant Green’s potentials on the unit ball in $\textbf {C}^ n$

Abstract:

Let $p(z) = \int _B {G(z, w) d\mu (w)}$ be an invariant Green’s potential on the unit ball $B$ in ${{\mathbf {C}}^n}\;(n \geqslant 1)$, where $G$ is the invariant Green’s function and $\mu$ is a positive measure with $\int _B {{{(1 - |w{|^2})}^n} d\mu (w) < \infty }$. In this paper, a necessary and sufficient condition on a subset $E$ of $B$ such that for every invariant Green’s potential $p$, \[ \lim \limits _{z \to e} \inf {(1 - |z{|^2})^n}p(z) = 0,\qquad e = (1, 0, \ldots , 0)\; \in \partial B,\;z \in E,\] is given. The condition is that the capacity of the sets $E \cap \{ z \in B|\;|z - e| < \varepsilon \}$, $\varepsilon > 0$, is bounded away from $0$. The result obtained here generalizes Luecking’s result, see [L], on the unit disc in ${\mathbf {C}}$.