Minimal harmonic functions on Denjoy domains

Abstract:

Let $\Omega = {\mathbb {R}^n}\backslash E$, where $E$ is a closed subset of the hyperplane $\left \{ {{x_n} = 0} \right \}$ and every point of $E$ is regular for the Dirichlet problem on $\Omega$. Further, let ${\alpha _k}$. denote the $(n - 1)$-dimensional measure of the set $\{ X \in \Omega :{x_n} = 0,{e^k} < |X| < {e^{k + 1}}\}$. It is known that the cone, ${\mathcal {P}_E}$, of positive harmonic functions on $\Omega$ which vanish on $E$ has dimension 1 or 2. In this paper it is shown that if $\sum {{e^{ - nk}}\alpha _k^{n/(n - 1)} < + \infty }$ then $\dim {\mathcal {P}_E} = 2$. This result, which in the case $n = 2$ implies a recent theorem of Segawa, is also shown to be sharp.