### 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 |

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 |

### Fingerprint

### 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

### Cite this

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

**Pseudodifferential operators on variable lebesgue spaces.** / Karlovich, Alexei Yu; Spitkovsky, Ilya.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*Operator Theory, Pseudo-Differential Equations, and Mathematical Physics.*Operator Theory: Advances and Applications, vol. 228, Springer International Publishing, pp. 173-183, International workshop on Analysis, Operator Theory, and Mathematical Physics, 2012, Ixtapa, Mexico, 1/23/12.

}

TY - GEN

T1 - Pseudodifferential operators on variable lebesgue spaces

AU - Karlovich, Alexei Yu

AU - Spitkovsky, Ilya

PY - 2013/1/1

Y1 - 2013/1/1

N2 - Let M(ℝn) be the class of bounded away from one and infinity functions p: ℝn → [1,∞] such that the Hardy-Littlewood maximal operator is bounded on the variable Lebesgue space Lp(⋅)(ℝn). We show that if a belongs to the Hörmander class Sn(ρ-1) ρ, δ with 0 < ρ ≤ 1, 0 ≤ δ < 1, then the pseudodifferential operator Op(a) is bounded on Lp(⋅)(ℝn) provided that p∈M(ℝn). Let M* (ℝn) be the class of variable exponents p∈M(ℝn) represented as 1/p(x) = θ/p0 + (1 – θ)/p1(x) where p0 ∈ (1,∞), θ ∈(0, 1), and p1 ∈M(ℝn). We prove that if a ∈ S0 1,0 slowly oscillates at infinity in the first variable, then the condition (Formula presented.) is sufficient for the Fredholmness of Op(a) on Lp(⋅)(ℝn) whenever p∈M* (ℝn). Both theorems generalize pioneering results by Rabinovich and Samko [24] obtained for globally log-Hölder continuous exponents p, constituting a proper subset of M* (ℝn).

AB - Let M(ℝn) be the class of bounded away from one and infinity functions p: ℝn → [1,∞] such that the Hardy-Littlewood maximal operator is bounded on the variable Lebesgue space Lp(⋅)(ℝn). We show that if a belongs to the Hörmander class Sn(ρ-1) ρ, δ with 0 < ρ ≤ 1, 0 ≤ δ < 1, then the pseudodifferential operator Op(a) is bounded on Lp(⋅)(ℝn) provided that p∈M(ℝn). Let M* (ℝn) be the class of variable exponents p∈M(ℝn) represented as 1/p(x) = θ/p0 + (1 – θ)/p1(x) where p0 ∈ (1,∞), θ ∈(0, 1), and p1 ∈M(ℝn). We prove that if a ∈ S0 1,0 slowly oscillates at infinity in the first variable, then the condition (Formula presented.) is sufficient for the Fredholmness of Op(a) on Lp(⋅)(ℝn) whenever p∈M* (ℝn). Both theorems generalize pioneering results by Rabinovich and Samko [24] obtained for globally log-Hölder continuous exponents p, constituting a proper subset of M* (ℝn).

KW - Fefferman-Stein sharp maximal operator

KW - Fredholmness

KW - Hardy-Littlewood maximal operator

KW - Hörmander symbol

KW - Pseudodifferential operator

KW - Slowly oscillating symbol

KW - Variable Lebesgue space

UR - http://www.scopus.com/inward/record.url?scp=84946043539&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84946043539&partnerID=8YFLogxK

M3 - Conference contribution

AN - SCOPUS:84946043539

SN - 9783034805360

T3 - Operator Theory: Advances and Applications

SP - 173

EP - 183

BT - Operator Theory, Pseudo-Differential Equations, and Mathematical Physics

PB - Springer International Publishing

ER -