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