# 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.

#### References

 [1] V.I. Smirnov, "A course of higher mathematics" , 5 , Addison-Wesley (1964) (Translated from Russian) MR0182690 MR0182688 MR0182687 MR0177069 MR0168707 Zbl 0122.29703 Zbl 0121.25904 Zbl 0118.28402 Zbl 0117.03404 [2] V.S. Vladimirov, "Equations of mathematical physics" , MIR (1984) (Translated from Russian) MR0764399 Zbl 0954.35001 Zbl 0652.35002 Zbl 0695.35001 Zbl 0699.35005 Zbl 0607.35001 Zbl 0506.35001 Zbl 0223.35002 Zbl 0231.35002 Zbl 0207.09101 [3] M.A. Krasnosel'skii, et al., "Integral operators in spaces of summable functions" , Noordhoff (1976) (Translated from Russian) Zbl 0312.47041