# Weak singularity

polar singularity

The unboundedness of an integral kernel $K ( x, s)$( cf. Kernel of an integral operator) when the product $M( x, s)= | x - s | ^ \alpha K ( x, s)$, $( x, s) \in \Omega \times \Omega$, is bounded. Here, $\Omega$ is a set in the space $\mathbf R ^ {n}$, $| x - s |$ is the distance between two points $x$ and $s$ and $0 < \alpha = \textrm{ const } < n$. The integral operator generated by such a kernel,

$$\tag{1 } K \phi ( t) = \int\limits _ \Omega \frac{M ( x, s) }{| x - s | ^ \alpha } \phi ( s) ds,$$

is called an integral operator with a weak singularity (or with a polar singularity). Let $\Omega$ be a compact subset of $\mathbf R ^ {n}$. If $M ( x, s)$ is continuous on $\Omega \times \Omega$, the operator (1) is completely continuous (cf. Completely-continuous operator) on the space of continuous functions $C ( \Omega )$, and if $M$ is bounded, then the operator (1) is completely continuous on the space $L _ {2} ( \Omega )$.

The kernel

$$\tag{2 } ( K _ {1} \otimes K _ {2} ) ( x, s) = \ \int\limits _ \Omega K _ {1} ( x, t) K _ {2} ( t, s) dt$$

is called the convolution of the kernels $K _ {1}$ and $K _ {2}$. Suppose $K _ {1} , K _ {2}$ have weak singularities, with

$$| K _ {i} ( x, s) | \leq \ \frac{\textrm{ const } }{| x- s | ^ {\alpha _ {i} } } ,\ \ \alpha _ {i} = \textrm{ const } < n,\ i = 1, 2;$$

then their convolution (2) is bounded or has a weak singularity, and:

 | K _ {1} \otimes K _ {2} ( x, s) | \leq \ \left \{

where $c$ is a constant.

If a kernel has a weak singularity, then all its iterated kernels from some iteration onwards are bounded.

How to Cite This Entry:
Weak singularity. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Weak_singularity&oldid=49183
This article was adapted from an original article by B.V. Khvedelidze (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article