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''polar singularity''

The unboundedness of an integral kernel  $  K ( x, s) $(
cf. [[Kernel of an integral operator|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|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====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> V.I. Smirnov, "A course of higher mathematics" , '''5''' , Addison-Wesley (1964) (Translated from Russian) {{MR|0182690}} {{MR|0182688}} {{MR|0182687}} {{MR|0177069}} {{MR|0168707}} {{ZBL|0122.29703}} {{ZBL|0121.25904}} {{ZBL|0118.28402}} {{ZBL|0117.03404}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> V.S. Vladimirov, "Equations of mathematical physics" , MIR (1984) (Translated from Russian) {{MR|0764399}} {{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}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> M.A. Krasnosel'skii, et al., "Integral operators in spaces of summable functions" , Noordhoff (1976) (Translated from Russian) {{MR|}} {{ZBL|0312.47041}} </TD></TR></table>

====Comments====
Weakly-singular kernels appear frequently in the boundary integral equation method for solving elliptic equations (see [[#References|[a1]]]). Another important integral equation with a weakly-singular kernel is the [[Abel integral equation|Abel integral equation]] ([[#References|[a2]]]).

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> D.L. Colton, R. Kress, "Integral equation methods in scattering theory" , Wiley (1983) {{MR|0700400}} {{ZBL|0522.35001}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> R. Gorenflo, S. Vessella, "Abel integral equations in analysis and applications" , Springer (1991)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> P.P. Zabreiko (ed.) A.I. Koshelev (ed.) M.A. Krasnoselskii (ed.) S.G. Mikhlin (ed.) L.S. Rakovshchik (ed.) V.Ya. Stet'senko (ed.) T.O. Shaposhnikova (ed.) R.S. Anderssen (ed.) , ''Integral equations - a reference text'' , Noordhoff (1975) pp. Sects. I.1.2; II.6 (Translated from Russian)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> H. Hochstadt, "Integral equations" , Wiley (1975) pp. Sect. II.4 {{MR|1013363}} {{MR|0390680}} {{MR|0190666}} {{ZBL|0718.45001}} {{ZBL|0259.45001}} {{ZBL|0137.08601}} </TD></TR></table>