### Abstract

Let M(ℝ^{n}) be the class of bounded away from one and infinity functions p: ℝ^{n} → [1,∞] such that the Hardy-Littlewood maximal operator is bounded on the variable Lebesgue space L^{p(⋅)}(ℝ^{n}). We show that if a belongs to the Hörmander class S^{n(ρ-1)} _{ρ, δ} with 0 < ρ ≤ 1, 0 ≤ δ < 1, then the pseudodifferential operator Op(a) is bounded on L^{p(⋅)}(ℝ^{n}) provided that p∈M(ℝ^{n}). Let M* (ℝ^{n}) be the class of variable exponents p∈M(ℝ^{n}) represented as 1/p(x) = θ/p_{0} + (1 – θ)/p_{1}(x) where p_{0} ∈ (1,∞), θ ∈(0, 1), and p_{1} ∈M(ℝ^{n}). We prove that if a ∈ S^{0} _{1,0} slowly oscillates at infinity in the first variable, then the condition (Formula presented.) is sufficient for the Fredholmness of Op(a) on L^{p(⋅)}(ℝ^{n}) whenever p∈M* (ℝ^{n}). Both theorems generalize pioneering results by Rabinovich and Samko [24] obtained for globally log-Hölder continuous exponents p, constituting a proper subset of M* (ℝ^{n}).

Original language | English (US) |
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Title of host publication | Operator Theory, Pseudo-Differential Equations, and Mathematical Physics |

Editors | Yuri I. Karlovich, Luigi Rodino, Ilya M. Spitkovsky, Bernd Silbermann |

Publisher | Springer International Publishing |

Pages | 173-183 |

Number of pages | 11 |

ISBN (Print) | 9783034805360 |

State | Published - Jan 1 2013 |

Event | International workshop on Analysis, Operator Theory, and Mathematical Physics, 2012 - Ixtapa, Mexico Duration: Jan 23 2012 → Jan 27 2012 |

### Publication series

Name | Operator Theory: Advances and Applications |
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Volume | 228 |

ISSN (Print) | 0255-0156 |

ISSN (Electronic) | 2296-4878 |

### Other

Other | International workshop on Analysis, Operator Theory, and Mathematical Physics, 2012 |
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Country | Mexico |

City | Ixtapa |

Period | 1/23/12 → 1/27/12 |

### Keywords

- Fefferman-Stein sharp maximal operator
- Fredholmness
- Hardy-Littlewood maximal operator
- Hörmander symbol
- Pseudodifferential operator
- Slowly oscillating symbol
- Variable Lebesgue space

### ASJC Scopus subject areas

- Analysis

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## Cite this

*Operator Theory, Pseudo-Differential Equations, and Mathematical Physics*(pp. 173-183). (Operator Theory: Advances and Applications; Vol. 228). Springer International Publishing.