J. M. Odinabekov

Lomonosov Moscow Statе University in Dushanbe (Dushanbe, Tajikistan)

Abstract. This work is devoted to study of two-dimensional singular integral operators with continuous coefficients defined on bounded domains of the complex plane. Such operators play an important role in solving a wide range of problems in mathematical physics, boundary value theory, and functional analysis. The relevance of the research is determined by need to develop effective methods for analyzing the properties of these operators, including questions of their solvability and the derivation of formulas for calculating their indices. The fundamentals of the general theory of multidimensional singular integral operators on unbounded domains were established in the works of S. G. Mikhlin. For two-dimensional operators with a non--zero symbol, the Fredholm theory plays a crucial role. When singular integral operators are considered on bounded domains, the solvability of the corresponding equations depends essentially on the geometric and analytic properties of the boundary. That's why this work investigates two-dimensional singular integral operators defined on bounded domains. To study this class of operators, the scheme proposed by R. Duduchava is used, which reduces the original problem to the analysis of an associated Riemann boundary value problem. The resulting Riemann problem is studied using the method developed by B. Bojarski. This approach is based on the construction of a special matrix function and on the splitting of set of polynomial matrices into homotopy classes. This approach makes it possible to investigate in detail the structure of the operators and their spectral characteristics. For the class of operators considered in Lebesgue spaces, effective criteria for their noethericity are established in the form of explicit conditions imposed on the operator coefficients. In addition, computational formulas for indices are obtained, making it possible to determine the main Fredholm properties of the operators and to investigate the conditions for existence and uniqueness of solutions to the corresponding integral equations.

Key Words: singular integral operator, operator index, operator symbol, Noethericity of operator

For citation: J. M. Odinabekov. Noethericity and computation of the index of two-dimensional singular integral operators in a bounded domain. Zhurnal Srednevolzhskogo matematicheskogo obshchestva. 28:2(2026), 11–27. DOI: https://doi.org/10.15507/2079-6900.28.202602.11-27

Information about the author:

Jasur M. Odinabekov, Ph.D. (Phys. and Math.), Head of Department of Mathematics and Natural Sciences, Lomonosov Moscow State University in Dushanbe (Tajikistan, Dushanbe, st. Bokhtar, 35/1), ORCID: http://orcid.org/0000-0003-3002-281X, jasur - 79@inbox.ru