---
tag: BMRX3X7095
kind: definition
name: Weakly singular kernel
msc: [45P05, 45H05]
defines: 94TNBAAJYX
uses: [GSMBXT7FPK, 0X3WGX95FE]
sources:
  - url: https://en.wikipedia.org/wiki/Integral_equation
---

## Definition

Let $m \ge 1$ and let $\Omega \subseteq \R^m$ be a closed bounded set.
Write $\abs{x - s}$ for the Euclidean distance of $x, s \in \Omega$.
A *weakly singular kernel* on $\Omega$ is a [[kernel|GSMBXT7FPK]] $K$ defined for $x, s \in \Omega$ with $x \ne s$ such that:

1. $K$ is a [[continuous function|0X3WGX95FE]] on $\set{(x, s) \in \Omega \times \Omega : x \ne s}$.
2. There are constants $M > 0$ and $\alpha \in [0, m)$ such that $\abs{K(x, s)} \le M \abs{x - s}^{-\alpha}$ for all $x \ne s$.

## Remarks

Example: $K(x, s) = \abs{x - s}^{-1/2}$ on $\Omega = [0, 1]$, with $m = 1$, $M = 1$ and $\alpha = 1/2 < 1$.
It is continuous off the diagonal.

Non-example: $K(x, s) = 1/(x - s)$ on $[0, 1]$.
It satisfies the bound only with $\alpha = 1$, which is not less than $m = 1$; the integral $\int_0^1 \abs{x - s}^{-1} \, ds$ diverges.

Since $\alpha < m$, the function $s \mapsto \abs{x - s}^{-\alpha}$ is integrable over $\Omega$, so $\int_\Omega K(x, s) \varphi(s) \, ds$ exists for bounded continuous $\varphi$.
Assume also that $\Omega$ is Jordan measurable, for example a closed interval or a closed ball.
Then the integral operator of a weakly singular kernel maps continuous functions on $\Omega$ to continuous functions and is compact for the maximum norm.
The kernel $\log \abs{x - s}$ is weakly singular on a bounded set: it satisfies the bound for every $\alpha \in (0, m)$.