An Algebra of Shift-invariant Singular Integral Operators with Slowly Oscillating Data and Its Application to Operators with a Carleman Shift

Abstract

The paper is devoted to studying Banach algebras of shift-invariant singular integral operators with slowly oscillating coefficients and their extensions by shift operators associated with iterations of a slowly oscillating Carleman shift generating a finite cyclic group. Both algebras are contained in the Banach algebra of bounded linear operators on a weighted Lebesgue space with a slowly oscillating Muckenhoupt weight over a composed slowly oscillating Carleson curve. By applying the theory of Mellin pseudodifferential operators, Fredholm symbol calculi for these algebras and Fredholm criteria and index formulas for their elements are established in terms of their Fredholm symbols.