Criterion of uniform invertibility of regular approximations of one-dimensional singular integral operators on a piecewise-Lyapunov contour

A. V. Abramyan ; V. S. Pilidi

Vladikavkazskij matematičeskij žurnal, Tome 21 (2019) no. 1, pp. 5-15

Résumé

The paper continues research of the criteria of applicability to complete singular integral operators of approximate methods using families of strongly approximating them operators with the “cut out” singularity of the Cauchy kernel. The case of a complete singular integral operator with continuous coefficients acting on $L_p$-space on a closed contour is considered. It is assumed that the contour is piecewise Lyapunov and has no cusps. The task is reduced to a criterion of invertibility of an element in some Banach algebra. The study is performed using the local principle of Gokhberg and Krupnik. The focus is on the local analysis at the corner points. For this purpose, an analogue of the method of quasi-equivalent operators proposed by I. B. Simonenko is used. The criterion is formulated in terms of invertibility of some integral operators associated with the corner points acting on $L_p$-space on the real axis, and strong ellipticity conditions at the contour points with the Lyapunov condition.