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

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 in [1, p. 73; 2, p. 176]; 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(Γ) of continuous functions, the Hölder space Hβ(Γ), 0 < β < 1, and the Lebesgue space Lp(Γ), 1 < p < ∞, for weakly singular integral operators and some modifications of such operators defined on arbitrary closed contours 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 β(Γ) and the Lebesgue space Lp(Γ), 1 < p < ∞, we theoretically justify the "kernel cuto" 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. © 2005 Pleiades Publishing, Inc.