Maximal Noncompactness of Singular Integral Operators on \(L^2\) Spaces with Some Khvedelidze Weights

Let \(\Gamma \) be a contour in the complex plane consisting of a finite number of circular arcs joining the endpoints \(-1\) and 1, possibly including the segment \([-1,1]\) . We consider the singular integral operator \(A=aI+bS_\Gamma \) with constant coefficients \(a,b\in \mathbb {C}\) , where \(S_\Gamma \) is the Cauchy singular integral operator over \(\Gamma \) . We provide a detailed proof of the maximal noncompactness of the operator A on \(L^2\) spaces with the Khvedelidze weights \(\varrho (t)=|t-1|{ }^\beta |t+1|{ }^{-\beta }\) satisfying \(-1<\beta <1\) . This result was announced by Naum Krupnik in 2010, but its proof has never been published.