The algebra of singular integral operators on a closed Carleson curve (Q1595691)

The algebra of singular integral operators generated by the singular operator \[ S\varphi= \frac{1}{\pi i}\int_{\Gamma}\frac{\varphi(\tau) d\tau}{\tau-t},\qquad t\in \Gamma \] over a closed contour \(\Gamma\) and operators of multiplication by piece-wise continuous functions, is well known; in the case when \(\Gamma\) is a smooth (piecewise-Lyapunov) curve, Fredholmness of operators from this algebra in the spaces \(L^p(\Gamma, \rho)\) with the Khvedelidze weight was studied by \textit{I. C. Gohberg} and \textit{N. Ja. Krupnik} [Math. USSR, Izv. 5 (1971), 955-979 (1972; Zbl 0248.47025)] and that of singular integral operators with general Muckenhoupt weights was investigated by \textit{I. Spitkovsky} [J. Funct. Anal. 105, 129-143 (1992; Zbl 0761.45001)]. Based on their approach in [Integral Equations Oper. Theory 22, 127-161 (1995; Zbl 0823.47027)], where the authors studied this algebra in the case when \(\Gamma\) admits points of logarithmic whirl, in the present paper they deal with the case of an arbitrary Carleson curve \(\Gamma\) and construct the corresponding symbolic calculus and obtain a criterion of Fredholmness in \(L^p(\Gamma)\) and a formula for the index in this general case. The proofs are outlined. Reviewer's remark: These results on Carleson curves, already more than 5 years old, are known to be generalized by the authors to the case of the spaces \(L^p(\Gamma, \rho)\) with Muckenhoupt weights \(\rho\), in the paper [Dokl. Math. 57, No. 2, 193-196 (1998; Zbl 0963.47032); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 359, No. 2, 151-154 (1998)]; see also authors' book [``Carleson curves, Muckenhoupt weights, and Toeplitz operators'' (Progress Math. 154, Birkhäuser) (1997; Zbl 0889.47001)] and their paper [Trans. Am. Math. Soc. 351, 3143-3196 (1999; Zbl 0918.47025)]; the paper by \textit{C. J.Bishop}, the authors and \textit{I. Spitkovsky} [Math. Nachr. 206, 5-83 (1999; Zbl 0982.47024)] and others are also relevant.