NONCOMPACTNESS OF FOURIER CONVOLUTION OPERATORS ON BANACH FUNCTION SPACES

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.