Estimates for weakly singular integral operators defined on closed integration contours and their applications to the approximate solution of singular integral equations.

*(English. Russian original)*Zbl 1128.45002
Differ. Equ. 41, No. 9, 1311-1322 (2005); translation from Differ. Uravn. 41, No. 9, 1242-1251 (2005).

Summary: To apply computational algorithms for the approximate solution of various classes of integral equations with weak singularities efficiently, one has to use certain “upper” bounds in Banach spaces for the weakly singular integral operators and their modifications occurring in these algorithms. Note that only the existence of constants in the upper bounds for weakly singular integral operators was proved by S. G. Mikhlin [Integral equations. (1949), p. 73] and by N. I. Muskhelishvili [Singular integral equations. 3rd ed. (1967; Zbl 0174.16201)]; however, this does not permit one to use computational algorithms in practice efficiently.

In the present paper, we compute the above-mentioned constants in the space \(C(\Gamma)\) of continuous functions, the Hölder space \(H_\beta(\Gamma)\), \(0 < \beta < 1\), and the Lebesgue space \(L_p(\Gamma)\), \(1 < p <\infty\), for weakly singular integral operators and some modifications of such operators defined on arbitrary closed contours \(\Gamma\) in the complex plane and establish invertibility conditions for the modified weakly singular integral operators in these spaces. The results will be essentially used in the theoretical justification of approximate methods for solving various classes of integral equations with weak singularities. In the Hölder space \(H_\beta(\Gamma)\) and the Lebesgue space \(L_p(\Gamma)\), \(1 < p < \infty\), we theoretically justify the “kernel cutoff” technique for the solution of singular integral equations that simultaneously contain the Carleman shift and the complex conjugate of the unknown function on arbitrary closed Lyapunov contours.

In the present paper, we compute the above-mentioned constants in the space \(C(\Gamma)\) of continuous functions, the Hölder space \(H_\beta(\Gamma)\), \(0 < \beta < 1\), and the Lebesgue space \(L_p(\Gamma)\), \(1 < p <\infty\), for weakly singular integral operators and some modifications of such operators defined on arbitrary closed contours \(\Gamma\) in the complex plane and establish invertibility conditions for the modified weakly singular integral operators in these spaces. The results will be essentially used in the theoretical justification of approximate methods for solving various classes of integral equations with weak singularities. In the Hölder space \(H_\beta(\Gamma)\) and the Lebesgue space \(L_p(\Gamma)\), \(1 < p < \infty\), we theoretically justify the “kernel cutoff” technique for the solution of singular integral equations that simultaneously contain the Carleman shift and the complex conjugate of the unknown function on arbitrary closed Lyapunov contours.

##### MSC:

45E10 | Integral equations of the convolution type (Abel, Picard, Toeplitz and Wiener-Hopf type) |

45P05 | Integral operators |

45L05 | Theoretical approximation of solutions to integral equations |

##### Keywords:

algorithms; kernel cutoff technique; singular integral operators; Carleman shift; Lyapunov contours##### Citations:

Zbl 0174.16201
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\textit{V. N. Seichuk}, Differ. Equ. 41, No. 9, 1311--1322 (2005; Zbl 1128.45002); translation from Differ. Uravn. 41, No. 9, 1242--1251 (2005)

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##### References:

[1] | Mikhlin, S.G., Integral’nye uravneniya (Integral Equations), Moscow, 1949. |

[2] | Muskhelishvili, N.I., Singulyarnye integral’nye uravneniya (Singular Integral Equations), Moscow, 1968. · Zbl 0174.16202 |

[3] | Seiciuc, V., Cattani, C., and Seiciuc, E., Analele ATIC-2003, Chisinau, Evrica, 2003, vol. 1(4), pp. 87–105. |

[4] | Lyusternik, L.A. and Sobolev, V.I., Elementy funktsional’nogo analiza (The Elements of Functional Analysis), Moscow, 1965. · Zbl 0141.11601 |

[5] | Seichuk, V.N., Trudy XI Mezhdunarodnogo simpoziuma ”MDOZMF-2003” (Proc. XI Int. Symp. ”MDOZMF-2003”), Kherson, 2003, pp. 277–281. |

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