Note on Fourier inequalities

Abstract

We prove that the Hausdorff--Young inequality $\Vert \hat{f}{\Vert }_{q(\cdot )}\le C\Vert f{\Vert }_{p(\cdot )}$ with $q(x)=p^{\prime }(1/x)$ 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.