---
tag: G6S707SYF9
kind: definition
name: Martin kernel
msc: [31C35, 31B05]
defines: 1DBC9FYF7M
uses: [M36SD56MWQ, 0DEB0JMN0W, TCYSWCTYYM, YNH832D50W]
sources:
  - url: https://en.wikipedia.org/wiki/Poisson_boundary
---

## Definition

Let $n \ge 2$ and let $\Omega \subseteq \R^n$ be a domain that has a [[Green's function|M36SD56MWQ]] $G \colon \Omega \times \Omega \to (0, +\infty]$.
This holds for every domain when $n \ge 3$, and for a plane domain whose complement is not a polar set.
Fix a reference point $x_0 \in \Omega$.
The *Martin kernel* of $\Omega$ with reference point $x_0$ is the function
$$M(x, y) = \frac{G(x, y)}{G(x_0, y)} \qquad \text{for } x \in \Omega,\ y \in \Omega \setminus \set{x_0}.$$

## Remarks

For fixed $y$, the function $M(\cdot, y)$ is positive and [[harmonic|0DEB0JMN0W]] on $\Omega \setminus \set{y}$, and $M(x_0, y) = 1$.
By [[Harnack's inequality|YNH832D50W]], when $y$ stays away from a compact set $K \ni x_0$, the functions $M(\cdot, y)$ are uniformly bounded on $K$.
Let $(y_k)$ be a sequence that leaves each compact subset of $\Omega$.
Then a subsequence of $M(\cdot, y_k)$ converges locally uniformly to a positive harmonic function $h$ with $h(x_0) = 1$.
Example: let $\Omega$ be the unit ball of $\R^n$ and $x_0 = 0$.
If $y_k \to \zeta$ with $\abs{\zeta} = 1$, then $M(x, y_k) \to (1 - \abs{x}^2) / \abs{x - \zeta}^n$, the [[Poisson kernel|TCYSWCTYYM]] for the normalised surface measure.
A change of reference point $x_0$ multiplies $M(\cdot, y)$ by the factor $G(x_0, y) / G(x_0', y)$.