Note on Fourier inequalities

We prove that the Hausdorff–Young inequality $\mathrm{\parallel }\widehat{f}{\mathrm{\parallel }}_{q(\cdot )}\le C\mathrm{\parallel }f{\mathrm{\parallel }}_{p(\cdot )}$ with $q(x)=p^{\mathrm{\prime }}(1\mathrm{/}x)$ and $p(\cdot )$ 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.