Boundary behavior of invariant Green’s potentials on the unit ball in 𝐶ⁿ

Abstract

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