### Abstract

We prove that if the Hardy-Littlewood maximal operator is bounded on a separable Banach function space X(R-n) and on its associate space X'(R-n) and a maximally modulated Calderon-Zygmund singular integral operator T-Phi is of weak type (r, r) for all r is an element of E (1, infinity), then T-Phi extends to a bounded operator on X(R-n). This theorem implies the boundedness of the maximally modulated Hilbert transform on variable Lebesgue spaces L-p((.)) (H) under natural assumptions on the variable exponent p : R -> (1, infinity). Applications of the above result to the boundedness and compactness of pseudodifferential operators with L-infinity, V(R))-symbols on variable Lebesgue spaces L-p((.)) (H) are considered. Here the Banach algebra L-infinity(R, V(R)) consists of all bounded measurable V(R)-valued functions on R where V(R) is the Banach algebra of all functions of bounded total variation.

Original language | English |
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Title of host publication | FUNCTION SPACES IN ANALYSIS |

Editors | K Jarosz |

Publisher | AMER MATHEMATICAL SOC |

Pages | 165-178 |

Number of pages | 14 |

ISBN (Print) | 978-1-4704-1694-2 |

DOIs | |

Publication status | Published - 2015 |

Event | 7th Conference on Function Spaces - Edwardsville, Israel Duration: 20 May 2014 → 24 May 2014 |

### Publication series

Name | Contemporary Mathematics |
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Publisher | AMER MATHEMATICAL SOC |

Volume | 645 |

ISSN (Print) | 0271-4132 |

### Conference

Conference | 7th Conference on Function Spaces |
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Country | Israel |

City | Edwardsville |

Period | 20/05/14 → 24/05/14 |

### Keywords

- Maximally modulated singular integral operator
- Calderon-Zygmund operator
- Hilbert transform
- pseudodifferential operator with non-regular symbol
- Banach function space
- variable Lebesgue space
- LEBESGUE SPACES
- BOUNDEDNESS
- COMPACTNESS
- SYMBOLS

## Cite this

*FUNCTION SPACES IN ANALYSIS*(pp. 165-178). (Contemporary Mathematics; Vol. 645). AMER MATHEMATICAL SOC. https://doi.org/10.1090/conm/645/1.2908