Let be a separable Banach function space such that the Hardy–Littlewood maximal operator is bounded on and on its associate space . Suppose that is a Fourier multiplier on the space . We show that the Fourier convolution operator with symbol is compact on the space if and only if . This result implies that nontrivial Fourier convolution operators on Lebesgue spaces with Muckenhoupt weights are never compact.
"Noncompactness of Fourier convolution operators on Banach function spaces." Ann. Funct. Anal. 10 (4) 553 - 561, November 2019. https://doi.org/10.1215/20088752-2019-0013