Invertibility of fourier convolution operators with piecewise continuous symbols on banach function spaces

We extend results on the invertibility of Fourier convolution operators with piecewise continuous symbols on the Lebesgue space Lp(R), p 2 (1;1), obtained by Roland Duduchava in the late 1970s, to the setting of a separable Banach function space X(R) such that the Hardy-Littlewood maximal operator is bounded on X(R) and on its associate space X0(R). We specify our results in the case of rearrangement-invariant spaces with suitable Muckenhoupt weights.