Maximal noncompactness of singular integral operators on \(L^p\) spaces with power weights (Q6611674)

The authors study maximal noncompactness of the singular integral operator \(A=a\, \mathrm{Id}+b\, S_{\Gamma}\) with constant coefficients \(a, \, b \in \mathbb{C}\). Here, \(S_{\Gamma}\) denotes the Cauchy singular integral operator over a curve \(\Gamma\) in the complex plane.\N\NThey extend \textit{N. Ya. Krupnik}'s results [Banach algebras with symbol and singular integral operators. Transl. from the Russian by A. Iacob. Birkhäuser, Cham (1987; Zbl 0641.47031); Oper. Theory: Adv. Appl. 202, 365--393 (2010; Zbl 1201.47046)]\N so as to include the Hausdorff measure of noncompactness. Assuming that \(\Gamma=\mathbb{R}\), they show that the above singular integral operator is maximally noncompact, i.e., the norm of the integral operator is equal to its essential norm and the norm of the integral operator is also equal to its Hausdorff measure of noncompactness at the same time.\N\NFor the entire collection see [Zbl 1544.35009].