Mellin Pseudodifferential Operators and Singular Integral Operators with Complex Conjugation

The paper is devoted to studying Banach algebras \({\mathfrak B}_{SO}\) and \({\mathfrak B}_{QC}\) of singular integral operators with complex conjugation and slowly oscillating (SO) or quasicontinuous (QC) coefficients on weighted Lebesgue spaces over star-like curves \(\Gamma \) without cusps by applying algebras of Mellin pseudodifferential operators with non-regular symbols. Making use of Mellin pseudodifferential operators with slowly oscillating \(V(\mathbb {R})\) -valued symbols of limited smoothness on \(\mathbb {R}_+\) or with quasicontinuous \(V(\mathbb {R})\) -valued symbols on \(\mathbb {R}_+\) , where \(V(\mathbb {R})\) is the Banach algebra of absolutely continuous functions of bounded total variation on \(\mathbb {R}\) , we construct a Fredholm symbol map for the Banach algebra \({\mathfrak B}_{SO}\) and obtain a Fredholm criterion and an index formula for the operators \(T\in {\mathfrak B}_{SO}\) , and construct a Fredholm symbol map for the Banach algebra \({\mathfrak B}_{QC}\) and establish a Fredholm criterion for the operators \(T\in {\mathfrak B}_{QC}\) .