### 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 language | Unknown |
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Title of host publication | Operator Theory Advances and Applications |

Editors | J Ball, V Bolotnikov, J Helton, L Rodman, I Spitkovsky |

Place of Publication | VIADUKSTRASSE 40-44, PO BOX 133, CH-4010 BASEL, SWITZERLAND |

Publisher | BIRKHAUSER VERLAG AG |

Pages | 321-336 |

Volume | 202 |

ISBN (Print) | 978-3-0346-0157-3 |

DOIs | |

Publication status | Published - 1 Jan 2010 |

Event | International Workshop on Operator Theory and Applications, IWOTA 2008 - Duration: 1 Jan 2008 → … |

### Conference

Conference | International Workshop on Operator Theory and Applications, IWOTA 2008 |
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Period | 1/01/08 → … |

## 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