### Abstract

We extend the main result of [12] to the case of more general weighted singular integral operators with two shifts of the form (aI - bU_{α})P_{γ} ^{+} + (cI - dU_{β})P_{γ} ^{-}, acting on the space L^{p}(ℝ_{+}), 1 < p < ∞, where P_{γ} ^{±} = (I ± S_{γ})/2 are operators associated with the weighted Cauchy singular integral operator S_{γ}, given by with γ εℂ satisfying 0 < 1/p + _{γ} < 1, and U_{α}, U_{β} are the isometric shift operators given by U_{α}f = (α^{'})^{1/p}(f ο α), U_{β}f = (β^{'})^{1/p}(f ο β), generated by diffeomorphisms α β of ℝ_{+} onto itself having only two βxed points at the endpoints 0 and ∞, under the assumptions that the coefficients a; b; c; d and the derivatives α^{'}, β^{'} of the shifts are bounded and continuous on ℝ_{+} and admit discontinuities of slowly oscillating type at 0 and ∞.

Original language | English |
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Pages (from-to) | 365-399 |

Number of pages | 35 |

Journal | Journal of Integral Equations and Applications |

Volume | 29 |

Issue number | 3 |

DOIs | |

Publication status | Published - 2017 |

### Keywords

- Fredholmness
- Orientation-preserving shift
- Slowly oscillating function
- Weighted Cauchy singular integral operator

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## Cite this

*Journal of Integral Equations and Applications*,

*29*(3), 365-399. https://doi.org/10.1216/JIE-2017-29-3-365