View source for Weak singularity

''polar singularity''

The unboundedness of an integral kernel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097230/w0972301.png" /> (cf. [[Kernel of an integral operator|Kernel of an integral operator]]) when the product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097230/w0972302.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097230/w0972303.png" />, is bounded. Here, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097230/w0972304.png" /> is a set in the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097230/w0972305.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097230/w0972306.png" /> is the distance between two points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097230/w0972307.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097230/w0972308.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097230/w0972309.png" />. The integral operator generated by such a kernel,

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097230/w09723010.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>

is called an integral operator with a weak singularity (or with a polar singularity). Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097230/w09723011.png" /> be a compact subset of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097230/w09723012.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097230/w09723013.png" /> is continuous on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097230/w09723014.png" />, the operator (1) is completely continuous (cf. [[Completely-continuous operator|Completely-continuous operator]]) on the space of continuous functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097230/w09723015.png" />, and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097230/w09723016.png" /> is bounded, then the operator (1) is completely continuous on the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097230/w09723017.png" />.

The kernel

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097230/w09723018.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>

is called the convolution of the kernels <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097230/w09723019.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097230/w09723020.png" />. Suppose <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097230/w09723021.png" /> have weak singularities, with

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097230/w09723022.png" /></td> </tr></table>

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

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097230/w09723023.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097230/w09723024.png" /> 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)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  V.S. Vladimirov,   "Equations of mathematical physics" , MIR  (1984)  (Translated from Russian)</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)</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)</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</TD></TR></table>