Note on Fourier inequalities

We prove that the Hausdorff--Young inequality \|f\|ₐ () C \|f\|₏ () with q (x) =p' (1/x) and p () even and non-decreasing holds in variable Lebesgue spaces if and only if p is a constant. However, under the additional condition on monotonicity of f, we obtain a full characterization of Pitt-type weighted Fourier inequalities in the classical and variable Lebesgue setting.