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Estimation of norms and conditions of Noetherian property of perturbed singular integral operators


Author: Petru Hariton Moloşnic
Degree:doctor of physics and mathematics
Speciality: 01.01.01 - Mathematical analysis
Year:2004
Scientific adviser: Vasile Neagu
doctor habilitat, professor, Moldova State University
Institution:
Scientific council:

Status

The thesis was presented on the 29 October, 2004
Approved by NCAA on the 23 December, 2004

Abstract

Adobe PDF document – 0.19 Mb / in romanian

Keywords

singular integral operators, noetherion operators, Lyapunov contour, algebra with symbol, essential norm, local principle

Summary

The dissertation is devoted to the following questions of the theory of singular integral operators: the calculation (and estimatin) of exact constants in theorems of M. Riss, B. Khvedelidze and I. Simonenko on the boundedness of operator with Cauchy kernel in different functional spaces, the application of these results to finding sufficient conditions of Noetherian properthy for systems of singular integral equations with bounded coefficients, the investigation of singular operators perturbed by some linear non-compact operators. The integration contour is assumed to have finitely many corner points.

To solve these problems in the work the local principle was adapted to the calculation of norms of operators of local type, whin made it possible instead of the norm of initial operator to calculate the norm of some simplest ”canonical” representatives at cach point of the contour. A rather wide class of functions that are weights relative to singular integrals in the space p L with weight was obtained. Basing on these results the estimates of essential norms for projecting M. Riss operators in the case of contour with corner points were found. It is proved that the essential norms of these operators depend on the space p L , on the weight and on the integration contour. A method is obtained that allows us to reduce the question about Noetherian property of a singular integral equation with bounded measurable coefficients in a Lesbegue space with weight to the investigation of Noetherian property of an analogous aquation in a space without weight. It is proved that for characteristic singular operators the Noetheran property is stable relative to their perturbation by some non-compact operators.

The methods and results of our dissertation can be used for further instigation of singular integral operators and algebras generated by such operators; for application of such operators in the theory of boundary value problems of mathematical physics; for solution of particular integral equations in applied mathematics and other.