Note on Fourier inequalities

Abstract

We prove that the Hausdorff-Young inequality parallel to(f) over cap parallel to(q(center dot)) <= C parallel to F parallel to(p(center dot)) with q(x) = p '(1/x) and p(center dot) 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 complete characterization of Pitt-type weighted Fourier inequalities in both the classical and variable Lebesgue setting.