### Abstract

We establish a symbol calculus for deciding whether singular integral operators with piecewise continuous coefficients are Fredholm on the Lebesgue space L^{p}(Γ, w) where 1 < p < ∞, Γ is a composed Carleson curve, and w is a Muckenhoupt weight in the class A_{p}(Γ). We also provide index formulas for the operators in the closed algebra of singular integral operators with piecewise continuous matrix-valued coefficients. Our main theorem is based upon three pillars: on the identification of the local spectrum of the Cauchy singular integral operator at the endpoints of simple Carleson arcs, on an appropriate "N projections theorem", and on results of geometric function theory pertaining to the problem of extending Carleson curves and Muckenhoupt weights.

Original language | English (US) |
---|---|

Pages (from-to) | 5-83 |

Number of pages | 79 |

Journal | Mathematische Nachrichten |

Volume | 206 |

DOIs | |

State | Published - Jan 1 1999 |

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### Keywords

- Ahlfors-David curve
- Carleson condition
- Geometric function theory
- Idempotent
- Index
- Muckenhoupt condition
- Singular integral operator
- Spectral theory

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Mathematische Nachrichten*,

*206*, 5-83. https://doi.org/10.1002/mana.19992060102

**Local spectra and index of singular integral operators with piecewise continuous coefficients on composed curves.** / Bishop, C. J.; Böttcher, A.; Karlovich, Yu I.; Spitkovsky, Ilya.

Research output: Contribution to journal › Article

*Mathematische Nachrichten*, vol. 206, pp. 5-83. https://doi.org/10.1002/mana.19992060102

}

TY - JOUR

T1 - Local spectra and index of singular integral operators with piecewise continuous coefficients on composed curves

AU - Bishop, C. J.

AU - Böttcher, A.

AU - Karlovich, Yu I.

AU - Spitkovsky, Ilya

PY - 1999/1/1

Y1 - 1999/1/1

N2 - We establish a symbol calculus for deciding whether singular integral operators with piecewise continuous coefficients are Fredholm on the Lebesgue space Lp(Γ, w) where 1 < p < ∞, Γ is a composed Carleson curve, and w is a Muckenhoupt weight in the class Ap(Γ). We also provide index formulas for the operators in the closed algebra of singular integral operators with piecewise continuous matrix-valued coefficients. Our main theorem is based upon three pillars: on the identification of the local spectrum of the Cauchy singular integral operator at the endpoints of simple Carleson arcs, on an appropriate "N projections theorem", and on results of geometric function theory pertaining to the problem of extending Carleson curves and Muckenhoupt weights.

AB - We establish a symbol calculus for deciding whether singular integral operators with piecewise continuous coefficients are Fredholm on the Lebesgue space Lp(Γ, w) where 1 < p < ∞, Γ is a composed Carleson curve, and w is a Muckenhoupt weight in the class Ap(Γ). We also provide index formulas for the operators in the closed algebra of singular integral operators with piecewise continuous matrix-valued coefficients. Our main theorem is based upon three pillars: on the identification of the local spectrum of the Cauchy singular integral operator at the endpoints of simple Carleson arcs, on an appropriate "N projections theorem", and on results of geometric function theory pertaining to the problem of extending Carleson curves and Muckenhoupt weights.

KW - Ahlfors-David curve

KW - Carleson condition

KW - Geometric function theory

KW - Idempotent

KW - Index

KW - Muckenhoupt condition

KW - Singular integral operator

KW - Spectral theory

UR - http://www.scopus.com/inward/record.url?scp=0040705368&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0040705368&partnerID=8YFLogxK

U2 - 10.1002/mana.19992060102

DO - 10.1002/mana.19992060102

M3 - Article

AN - SCOPUS:0040705368

VL - 206

SP - 5

EP - 83

JO - Mathematische Nachrichten

JF - Mathematische Nachrichten

SN - 0025-584X

ER -