Geodesic
Inversion of integral operators with kernels discontinuous on the diagonal
Matematičeskie zametki, Tome 64 (1998) no. 6, pp. 932-942.

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Conditions implying the invertibility of the integral operator $$ Af(x)=\int_0^1A(x,t)f(t)\,dt $$ with kernel $A(x,t)$ having discontinuities of the first kind at the points $t=x$ and $t=1-x$ are found. We give explicit inversion formulas as well as applications to the problem of finding the square roots of the operator $y''(x)$ with arbitrary boundary conditions and the problem of expansion with respect to eigenfunctions. 
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     author = {A. P. Khromov},
     title = {Inversion of integral operators with kernels discontinuous on the diagonal},
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A. P. Khromov. Inversion of integral operators with kernels discontinuous on the diagonal. Matematičeskie zametki, Tome 64 (1998) no. 6, pp. 932-942. http://geodesic.mathdoc.fr/item/MZM_1998_64_6_a13/

[1] Khromov A. P., “Teoremy ravnoskhodimosti dlya integro-differentsialnykh i integralnykh operatorov”, Matem. sb., 114(156):3 (1981), 378–405 | MR | Zbl

[2] Khromov A. P., Gurevich A. P., “Teoremy ravnoskhodimosti dlya odnogo klassa integralnykh operatorov”, Vosmaya Saratovskaya zimnyaya shkola, Tezisy dokl., Saratov, 1996, 117

[3] Khromov A. P., “Ob obraschenii odnogo klassa integralnykh operatorov”, Vosmaya Saratovskaya zimnyaya shkola, Tezisy dokl., Saratov, 1996, 118