Cláudio A. Fernandes, Alexei Y. Karlovich, Yuri I. Karlovich. Source: Annals of Functional Analysis, Volume 10, Number 4, 553--561.Abstract:
Let $X(\mathbb{R})$ be a separable Banach function space such that the Hardy–Littlewood maximal operator $M$ is bounded on $X(\mathbb{R})$ and on its associate space $X'(\mathbb{R})$ . Suppose that $a$ is a Fourier multiplier on the space $X(\mathbb{R})$ . We show that the Fourier convolution operator $W^{0}(a)$ with symbol $a$ is compact on the space $X(\mathbb{R})$ if and only if $a=0$ . This result implies that nontrivial Fourier convolution operators on Lebesgue spaces with Muckenhoupt weights are never compact.