@inbook { ISI:000185212300009,
	title = {Compactness of commutators arising in the Fredholm theory of singular integral operators with shifts},
	booktitle = {Factorization, Singular Operators and Related Problems},
	year = {2003},
	pages = {111-129},
	publisher = {Kluwer Academic Publishers},
	organization = {Kluwer Academic Publishers},
	type = {{Proceedings Paper}},
	address = {Dordrecht},
	abstract = {<script src={\textquoteright}https://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML{\textquoteright}></script><p>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.</p>
},
	keywords = {Boyd indices, commutator, compact operator, interpolation of compactness}, rearrangement-invariant space, shift operator, {Cauchy singular integral operator},
	isbn = {{1-4020-1407-4}},
	url = {http://link.springer.com/chapter/10.1007/978-94-017-0227-0_9},
	author = {Karlovich, Alexei Yu. and Karlovich, Yuri I.},
	editor = {Samko, Stefan and Lebre, Amarino and Ferreira dos Santos, Ant{\'o}nio}
}
