Noncompactness of Fourier Convolution Operators on Banach Function Spaces

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 $X^{\mathrm{\prime }}(\mathbb{R})$. Suppose $a$ 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.