Maximal Noncompactness of Singular Integral Operators on \(L^p\) Spaces with Power Weights

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 a curve $$\Gamma $$ in the complex plane. We discuss two situations when the operator A is maximally noncompact on the space $$L^p(\Gamma ,\varrho )$$ with $$1