### Abstract

Let a be a semi-almost periodic matrix function with the almost periodic representatives al and ar at -∞ and +∞, respectively. Suppose p:R→(1,∞) is a slowly oscillating exponent such that the Cauchy singular integral operator S is bounded on the variable Lebesgue space Lp(.)(R). We prove that if the operator aP+Q with P=(I+S)/2 and Q=(I-S)/2 is Fredholm on the variable Lebesgue space LNp(.)(R), then the operators alP+Q and arP+Q are invertible on standard Lebesgue spaces LNql(R) and LNqr(R) with some exponents ql and qr lying in the segments between the lower and the upper limits of p at -∞ and +∞, respectively.

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

Pages (from-to) | 706-725 |

Number of pages | 20 |

Journal | Journal of Mathematical Analysis and Applications |

Volume | 384 |

Issue number | 2 |

DOIs | |

State | Published - Dec 15 2011 |

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### Keywords

- Almost-periodic function
- Fredholmness
- Invertibility
- Semi-almost periodic function
- Singular integral operator
- Slowly oscillating function
- Variable Lebesgue space

### ASJC Scopus subject areas

- Analysis
- Applied Mathematics

### Cite this

*Journal of Mathematical Analysis and Applications*,

*384*(2), 706-725. https://doi.org/10.1016/j.jmaa.2011.06.066

**On singular integral operators with semi-almost periodic coefficients on variable Lebesgue spaces.** / Karlovich, Alexei Yu; Spitkovsky, Ilya.

Research output: Contribution to journal › Article

*Journal of Mathematical Analysis and Applications*, vol. 384, no. 2, pp. 706-725. https://doi.org/10.1016/j.jmaa.2011.06.066

}

TY - JOUR

T1 - On singular integral operators with semi-almost periodic coefficients on variable Lebesgue spaces

AU - Karlovich, Alexei Yu

AU - Spitkovsky, Ilya

PY - 2011/12/15

Y1 - 2011/12/15

N2 - Let a be a semi-almost periodic matrix function with the almost periodic representatives al and ar at -∞ and +∞, respectively. Suppose p:R→(1,∞) is a slowly oscillating exponent such that the Cauchy singular integral operator S is bounded on the variable Lebesgue space Lp(.)(R). We prove that if the operator aP+Q with P=(I+S)/2 and Q=(I-S)/2 is Fredholm on the variable Lebesgue space LNp(.)(R), then the operators alP+Q and arP+Q are invertible on standard Lebesgue spaces LNql(R) and LNqr(R) with some exponents ql and qr lying in the segments between the lower and the upper limits of p at -∞ and +∞, respectively.

AB - Let a be a semi-almost periodic matrix function with the almost periodic representatives al and ar at -∞ and +∞, respectively. Suppose p:R→(1,∞) is a slowly oscillating exponent such that the Cauchy singular integral operator S is bounded on the variable Lebesgue space Lp(.)(R). We prove that if the operator aP+Q with P=(I+S)/2 and Q=(I-S)/2 is Fredholm on the variable Lebesgue space LNp(.)(R), then the operators alP+Q and arP+Q are invertible on standard Lebesgue spaces LNql(R) and LNqr(R) with some exponents ql and qr lying in the segments between the lower and the upper limits of p at -∞ and +∞, respectively.

KW - Almost-periodic function

KW - Fredholmness

KW - Invertibility

KW - Semi-almost periodic function

KW - Singular integral operator

KW - Slowly oscillating function

KW - Variable Lebesgue space

UR - http://www.scopus.com/inward/record.url?scp=79961020530&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=79961020530&partnerID=8YFLogxK

U2 - 10.1016/j.jmaa.2011.06.066

DO - 10.1016/j.jmaa.2011.06.066

M3 - Article

VL - 384

SP - 706

EP - 725

JO - Journal of Mathematical Analysis and Applications

JF - Journal of Mathematical Analysis and Applications

SN - 0022-247X

IS - 2

ER -