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

TL;DR

This paper establishes Fourier inequalities in weighted $L_p$ spaces involving Hardy-Cesàro and Hardy-Bellman operators, extending results to product Hardy spaces and analyzing operator boundedness across various functional spaces.

Contribution

It introduces new Fourier inequalities with Hardy-type operators in weighted spaces and extends these to product Hardy spaces for $p \,\leq\, 1$, also discussing operator boundedness.

Findings

Fourier inequalities in weighted $L_p$ spaces involving Hardy operators.

Extension of results to product Hardy spaces for $p\leq 1$.

Boundedness of Hardy operators in Lebesgue, Hardy, and BMO spaces.

Abstract

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}}$.