### Abstract

Let α,β be orientation-preserving homeomorphisms of [0,∞] onto itself, which have only two fixed points at 0 and ∞, and whose restrictions to R_{+} = (0, ∞) are diffeomorphisms, and let U_{α},U_{β} be the corresponding isometric shift operators on the space L^{p}(R_{+}), 1 < p 1 < ∞, given by (Formula presented.). We prove criteria for the n-normality and d-normality on the space L^{p}(R_{+}) of singular integral operators of the form A+P+_{γ} + A−P_{γ}, where (Formula presented.) are operators in the Wiener algebras of functional operators with shifts and the operators (Formula presented.) are associated to the weighted Cauchy singular integral operator (Formula presented.) where γ ∈ C satisfies the condition 0 < 1/p + γ < 1. We assume that the coefficients a_{k}, b_{k} for k ∈ Z and the derivatives of the shifts α',β' are bounded continuous functions on R_{+} which may have slowly oscillating discontinuities at 0 and ∞.

Original language | English |
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Pages (from-to) | 997-1027 |

Number of pages | 31 |

Journal | Proceedings of the London Mathematical Society |

Volume | 116 |

Issue number | 4 |

DOIs | |

Publication status | Published - 1 Apr 2018 |

### Keywords

- 45E05
- 47A53
- 47B33
- 47B35
- 47G10 (primary)
- 47G30 (secondary)

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## Cite this

*Proceedings of the London Mathematical Society*,

*116*(4), 997-1027. https://doi.org/10.1112/plms.12108