Compactness of Commutators Arising in the Fredholm Theory of Singular Integral Operators with Shifts

Abstract

The paper is devoted to the compactness of the commutators aS Г — S Г aI and W α S Г — S Г W α , where S Г is the Cauchy singular integral operator, a is a bounded measurable function, W α is the shift operator given by W α f = f o α, and α is a bi-Lipschitz homeomorphism (shift). The cases of the unit circle and the unit interval are considered. We prove that these commutators are compact on rearrangement-invariant spaces with nontrivial Boyd indices if and only if the function a or, respectively. the derivative of the shift a has vanishing mean oscillation.