On the symbol of singular operators in the case of contour with corner points

Abstract

This paper proposes a method for constructing a symbol for singular integral operators in the case of a piecewise Lyapunov contour. The definition of the symbol function involves numbers that characterize the space in which the research is being carried out, as well as the values of the corner points of the contour, which makes it possible to obtain formulas for calculating the essential norms of singular operators and conditions for the solvability of singular equations with a shift and complex conjugation. In obtaining these results, we will essentially rely on the well-known results of I. Gelfand concerning maximal ideals of commutative Banach algebras \cite{GRS}. In the absence of corner points on the integration contour, the results of this work are consistent with the results from [1].