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

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Factorization, Singular Operators and Related Problems

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.

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Author information

Authors and Affiliations

  1. Instituto Superior Técnico, Portugal

    Alexei Karlovich

  2. Universidad Autónoma del Estado de Morelos, México

    Yuri Karlovich

Authors
  1. Alexei Karlovich
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  2. Yuri Karlovich
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Editor information

Editors and Affiliations

  1. Universidade do Algarve, Faro, Portugal

    Stefan Samko

  2. Instituto Superior Técnico, Lisboa, Portugal

    Amarino Lebre  & António F. dos Santos  & 

Additional information

To Professor G. S. Litvinchuk on the occasion of his 70th birthday

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Karlovich, A., Karlovich, Y. (2003). Compactness of Commutators Arising in the Fredholm Theory of Singular Integral Operators with Shifts. In: Samko, S., Lebre, A., dos Santos, A.F. (eds) Factorization, Singular Operators and Related Problems. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-0227-0_9

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  • DOI: https://doi.org/10.1007/978-94-017-0227-0_9

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-90-481-6333-5

  • Online ISBN: 978-94-017-0227-0

  • eBook Packages: Springer Book Archive

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