  We obtain Fourier inequalities in the weighted $L_p$ spaces for any
$1<p<\infty$ involving the Hardy-Ces\`aro and Hardy-Bellman operators. We
extend these results to product Hardy spaces for $p\le 1$. Moreover,
boundedness of the Hardy-Ces\`aro and Hardy-Bellman operators in various spaces
(Lebesgue, Hardy, BMO) is discussed. One of our main tools is an appropriate
version of the Hardy-Littlewood-Paley inequality $ \|\widehat{f}\|_{L_{p',q}}
\lesssim \left\|f\right\|_{L_{p,q}}$.
