Singular Integral Operators on Variable Lebesgue Spaces over Arbitrary Carleson Curves

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Abstract

In 1968, Israel Gohberg and Naum Krupnik discovered that local spectra of singular integral operators with piecewise continuous coefficients on Lebesgue spaces L-p(Gamma) over Lyapunov curves have the shape of circular arcs. About 25 years later, Albrecht Bottcher and Yuri Karlovich realized that these circular arcs metamorphose to so-called logarithmic leaves with a median separating point when Lyapunov curves metamorphose to arbitrary Carleson curves. We show that this result remains valid in a more general setting of variable Lebesgue spaces L-p(.) (Gamma) where p : Gamma -> (1, infinity) satisfies the Dini-Lipschitz condition. One of the main ingredients of the proof is a new condition for the boundedness of the Cauchy singular integral operator on variable Lebesgue spaces with weights related to oscillations of Carleson curves.
Original languageUnknown
Title of host publicationOperator Theory Advances and Applications
EditorsJ Ball, V Bolotnikov, J Helton, L Rodman, I Spitkovsky
Place of PublicationVIADUKSTRASSE 40-44, PO BOX 133, CH-4010 BASEL, SWITZERLAND
PublisherBIRKHAUSER VERLAG AG
Pages321-336
Volume202
ISBN (Print)978-3-0346-0157-3
DOIs
Publication statusPublished - 1 Jan 2010
EventInternational Workshop on Operator Theory and Applications, IWOTA 2008 -
Duration: 1 Jan 2008 → …

Conference

ConferenceInternational Workshop on Operator Theory and Applications, IWOTA 2008
Period1/01/08 → …

Cite this

Karlovych, O. (2010). Singular Integral Operators on Variable Lebesgue Spaces over Arbitrary Carleson Curves. In J. Ball, V. Bolotnikov, J. Helton, L. Rodman, & I. Spitkovsky (Eds.), Operator Theory Advances and Applications (Vol. 202, pp. 321-336). BIRKHAUSER VERLAG AG. https://doi.org/10.1007/978-3-0346-0158-0_18