Hardy-Littlewood-type theorems for Fourier transforms in ${\mathbb{R}}^d$

Abstract

We obtain Fourier inequalities in the weighted $L_p$ spaces for any $1<p<\infty$ involving the Hardy-Cesàro and Hardy-Bellman operators. We extend these results to product Hardy spaces for $p\le 1$. Moreover, boundedness of the Hardy-Cesàro 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 $\Vert \hat{f}{\Vert }_{L_{p^{\prime },q}}\lesssim {\Vert f\Vert }_{L_{p,q}}$.