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

Comments

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

References

[a1]	D.L. Colton, R. Kress, "Integral equation methods in scattering theory" , Wiley (1983) MR0700400 Zbl 0522.35001
[a2]	R. Gorenflo, S. Vessella, "Abel integral equations in analysis and applications" , Springer (1991)
[a3]	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)
[a4]	H. Hochstadt, "Integral equations" , Wiley (1975) pp. Sect. II.4 MR1013363 MR0390680 MR0190666 Zbl 0718.45001 Zbl 0259.45001 Zbl 0137.08601