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Singular Integral Operators on Variable Lebesgue Spaces over Arbitrary Carleson Curves

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Topics in Operator Theory

Part of the book series: Operator Theory: Advances and Applications ((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.

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Authors and Affiliations

  1. Departamento de Matemática, Faculdade de Ciênciase Tecnologia, Universidade Nova de Lisboa, Quinta da Torre, 2829-516, Caparica, Portugal

    Alexei Yu. Karlovich

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  1. Alexei Yu. Karlovich
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Additional information

To Professor Israel Gohberg on the occasion of his eightieth birthday

Communicated by I.M. Spitkovsky.

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Karlovich, A.Y. (2010). Singular Integral Operators on Variable Lebesgue Spaces over Arbitrary Carleson Curves. In: Topics in Operator Theory. Operator Theory: Advances and Applications, vol 202. Birkhäuser Basel. https://doi.org/10.1007/978-3-0346-0158-0_18

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