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

Alexei Karlovich () and Yuri Karlovich ()
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Alexei Karlovich: Instituto Superior Técnico
Yuri Karlovich: Universidad Autónoma del Estado de Morelos

A chapter in Factorization, Singular Operators and Related Problems, 2003, pp 111-129 from Springer

Abstract: 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.

Keywords: Cauchy singular integral operator; shift operator; commutator; compact operator; rearrangement-invariant space; Boyd indices; interpolation of compactness. (search for similar items in EconPapers)
Date: 2003
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-94-017-0227-0_9

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DOI: 10.1007/978-94-017-0227-0_9

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