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Singular Integral Operators on Variable Lebesgue Spaces over Arbitrary Carleson Curves

  • Alexei Yu. Karlovich
Part of the Operator Theory: Advances and Applications book series (OT, volume 202)

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(Γ) over Lyapunov curves have the shape of circular arcs. About 25 years later, Albrecht Böttcher 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(·) (Γ) where p:GΓ→(1,∞) 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.

Keywords

Fredholmness variable Lebesgue space Dini-Lipschitz condition Carleson curve singular integral operator piecewise continuous coefficient spirality indices 

Mathematics Subject Classification (2000).

Primary 47B35 Secondary 45E05 46E30 47A68 

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Copyright information

© Birkhäuser Verlag Basel/Switzerland 2010

Authors and Affiliations

  • Alexei Yu. Karlovich
    • 1
  1. 1.Departamento de Matemática, Faculdade de Ciênciase TecnologiaUniversidade Nova de LisboaQuinta da TorreCaparicaPortugal

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