### Abstract

Let X(R) be a separable Banach function space such that the Hardy-Littlewood maximal operator M is bounded on X(R) and on its associate space X' (R). Suppose that a is a Fourier multiplier on the space X(R) We show that the Fourier convolution operator W-0(a) with symbol a is compact on the space X(R) if and only if a = 0. This result implies that nontrivial Fourier convolution operators on Lebesgue spaces with Muckenhoupt weights are never compact.

Original language | English |
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Pages (from-to) | 553-561 |

Journal | Annals of functional analysis |

Volume | 10 |

Issue number | 4 |

DOIs | |

Publication status | Published - Nov 2019 |

### Keywords

- Fourier convolution operator
- Compactness
- Banach function space
- Hardy-Littlewood maximal operator
- Lebesgue space with Muckenhoupt weight

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## Cite this

Fernandes, C. A., Karlovich, A., & Karlovich, Y. I. (2019). Noncompactness of Fourier convolution operators on Banach function spaces.

*Annals of functional analysis*,*10*(4), 553-561. https://doi.org/10.1215/20088752-2019-0013