Boundary behavior of Green potentials

Abstract:

A Green potential on the unit disk $\{ \left | z \right | < 1\}$ is a function $u(z)$ of the form \[ u(z) = \int {\log \left | {\frac {{1 - \bar wz}}{{z - w}}} \right |{\text { }}} d\alpha (w),\] where $\alpha$ is a positive measure such that $\smallint (1 - \left | w \right |)d\alpha (w)$ is finite. In this note I give a necessary and sufficient condition on a relatively closed subset $F$ of the unit disk in order that, for all such $u(z)$, \[ \liminf _{F \ni z \to 1} (1 - \left | z \right |) u(z) = 0. \] The condition is that the hyperbolic capacity of the portion of $F$ in arbitrarily small neighborhoods of 1 is bounded away from zero.