## Compactness of commutators arising in the Fredholm theory of singular integral operators with shifts

Citation:
Karlovich, Alexei Yu., and Yuri I. Karlovich. "Compactness of commutators arising in the Fredholm theory of singular integral operators with shifts." Factorization, Singular Operators and Related Problems. Eds. Stefan Samko, Amarino Lebre, and António Ferreira dos Santos. Dordrecht: Kluwer Academic Publishers, 2003. 111-129.

### Abstract:

The paper is devoted to the compactness of the commutators $$aS_\Gamma - S_\Gamma aI$$ and $$W_\alpha S_\Gamma - S_\Gamma W_\alpha$$, where $$S_\Gamma$$ is the Cauchy singular integral operator, $$a$$ is a bounded measurable function, $$W_\alpha$$ is the shift operator given by $$W_\alpha f = f\circ\alpha$$, and $$\alpha$$ 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.