On the noethericity conditions and the index of some two-dimensional singular integral operators

The main problems in the theory of singular integral operators are the problems of boundedness, invertibility, Noethericity, and calculation of the index. The general theory of multidimensional singular integral operators over the entire space $E_n$ was constructed by S. G. Mikhlin. It is known that in the two-dimensional case, if the symbol of an operator does not vanish, then the Fredholm theory holds. For operators over a bounded domain, the boundary of this domain significantly affects the solvability of the corresponding operator equations. In this paper, we consider two-dimensional singular integral operators with continuous coefficients over a bounded domain. Such operators are used in many problems in the theory of partial differential equations. In this regard, it is of interest to establish criteria for the considered operators to be Noetherian in the form of explicit conditions on their coefficients. The paper establishes effective necessary and sufficient conditions for two-dimensional singular integral operators to be Noetherian in Lebesgue spaces $L_{p}(D)$ (considered over the field of real numbers), $1<p<\infty ,$ and formulas for calculating indices are given. The method developed by R. V. Duduchava [Duduchava R. On multidimensional singular integral operators. I: The half-space case; II: The case of compact manifolds // J. Operator Theory, 1984, v. 11, 41–76 (I); 199–214 (II)]. In this case, the study of the Noetherian properties of operators is reduced to the factorization of the corresponding matrix-functions and finding their partial indices.