Green function with a pole at infinity

Let $K\subseteq \mathbb{C}$ be compact and nonpolar, meaning that $\mathrm{cap}(K)>0$ for the logarithmic capacity of Robin constant and logarithmic capacity of a compact set; by Capacity-polar sets, quasi-everywhere, and subharmonic polar sets this is exactly the statement that $K$ is not capacity-polar. Such a $K$ is nonempty, the complement $\mathbb{C}\setminus K$ is a nonempty open set, and it has exactly one unbounded connected component (The complement of a compact plane set has exactly one unbounded connected component); that component is written

$\mathrm{\Omega }:=\mathrm{\Omega }(K)\subseteq \mathbb{C}\setminus K\mathrm{.}$

By A complex domain is a nonempty connected open subset of $\mathbb{C}$, $\mathrm{\Omega }$ is a complex domain, and $\mathrm{\Omega }\mathrm{\ne }\mathbb{C}$ because $K\mathrm{\ne }\mathrm{\varnothing }$ is disjoint from it. It is called the exterior domain of $K$. Its boundary $\mathrm{\partial }\mathrm{\Omega }$ (interior, closure and boundary in the sense of Interior, closure, boundary, limit point, isolated point and dense subset of a metric space) is a compact subset of $K$: indeed $\mathrm{\partial }\mathrm{\Omega }\subseteq \mathrm{\partial }(\mathbb{C}\setminus K)\subseteq K$, and it is called the outer boundary of $K$. Every point of $\mathrm{\partial }\mathrm{\Omega }$ is a finite point of $\mathbb{C}$; the point $\mathrm{\infty }$ is not part of it.

Since $\mathrm{cap}(K)>0$, the Robin constant $V_K={\inf }_{\mu \in P(K)}I(\mu )$ of $K$ is a real number, $V_K<+\mathrm{\infty }$ (Robin constant and logarithmic capacity of a compact set); it is the Robin constant in the pole at infinity. Harmonicity below is that of Plane harmonic functions, and moduli are those of Real and imaginary parts, complex conjugation, and modulus.

A Green function of $\mathrm{\Omega }$ with pole at infinity and Robin constant $V_K$ is a function $g:\mathrm{\Omega }\to \mathbb{R}$ with the following four properties.

1. Positive and harmonic. $g(z)>0$ for every $z\in \mathrm{\Omega }$, and $g$ is harmonic on $\mathrm{\Omega }$.

2. Logarithmic normalization at infinity. Writing $\log$ for the natural logarithm and $\mathrm{\mid }\cdot \mathrm{\mid }$ for the modulus,

   $g(z)-\log \mathrm{\mid }z\mathrm{\mid }⟶V_K\text{as }\mathrm{\mid }z\mathrm{\mid }\to \mathrm{\infty },z\in \mathrm{\Omega },$

   meaning: for every real $\epsilon >0$ there is a real $r$ such that $\mathrm{\mid }g(z)-\log \mathrm{\mid }z\mathrm{\mid }-V_K\mathrm{\mid }<\epsilon$ for every $z\in \mathrm{\Omega }$ with $\mathrm{\mid }z\mathrm{\mid }>r$. Because $\mathrm{\Omega }$ is unbounded, this condition is never vacuous.

3. Local boundedness near finite boundary points. For every $\xi \in \mathrm{\partial }\mathrm{\Omega }$ there is a real $r>0$ with

   $\sup \{g(z):z\in \mathrm{\Omega },\mathrm{\mid }z-\xi \mathrm{\mid }<r\}<+\mathrm{\infty }\mathrm{.}$

4. Zero boundary limit quasi-everywhere. There is a Borel capacity-polar set $E\subseteq \mathrm{\partial }\mathrm{\Omega }$ such that

   ${\lim }_{\mathrm{\Omega }\ni z\to \xi }g(z)=0\text{for every }\xi \in \mathrm{\partial }\mathrm{\Omega }\setminus E\mathrm{.}$

   That is, $g$ has boundary limit $0$ quasi-everywhere on the outer boundary, in the sense of Capacity-polar sets, quasi-everywhere, and subharmonic polar sets applied to the compact conductor $\mathrm{\partial }\mathrm{\Omega }$.

Notation. If a Green function with pole at infinity and Robin constant $V_K$ exists and is unique under properties 1–4, that unique function is written

$z\mapsto g_{\mathrm{\Omega }}(z,\mathrm{\infty }),\text{equivalently }z\mapsto g_{\mathrm{\Omega }(K)}(z,\mathrm{\infty })\mathrm{.}$

Neither existence nor uniqueness is asserted by this definition: they are conclusions of the theorem that builds the Green function from the equilibrium potential, and it is only there that the notation $g_{\mathrm{\Omega }}(\cdot ,\mathrm{\infty })$ is licensed.

Independence from the finite-pole kernel. Properties 1–4 are stated for an unbounded exterior domain and normalize the logarithmic term with coefficient $+1$ and the additive constant $V_K$ at infinity. This is not the published finite-pole canonical Green kernel $z\mapsto g_{\mathrm{\Omega }}(z,a)$ of The canonical Green kernel of a plane domain, whose pole is a point $a\in \mathrm{\Omega }$ and whose local form is $g_{\mathrm{\Omega }}(z,a)=-\log \mathrm{\mid }z-a\mathrm{\mid }+h(z)$ with $h$ harmonic across $a$. No clause above is imported from, or reduces to, that kernel; the two are different objects even on a common domain.

Local boundedness in the comparison argument. Property 3 records the local upper bound at each finite boundary point used in the comparison argument that identifies two candidates. Property 4 is a boundary condition on the candidate, not a regularity assumption on $\mathrm{\partial }\mathrm{\Omega }$.

Why the boundary limit is only quasi-everywhere. For a general compact nonpolar $K$ the outer boundary can contain irregular points, at which the equilibrium potential need not tend to its boundary value; those points form a capacity-polar set. Requiring the limit $0$ at every boundary point would exclude the model function $g=V_K-U^{{\mu }_K}$ and would not be the convention under which existence holds. The quasi-everywhere convention is the one under which the Green function is characterized by properties 1–4.