Sharp weak bounds for the Hardy-Littlewood-P\'{o}lya operator and the weak bounds for the multilinear integral operator

TL;DR

This paper establishes sharp bounds for the Hardy-Littlewood-Pólya operator and weak bounds for a multilinear integral operator, advancing understanding of their behavior on weighted Lebesgue spaces.

Contribution

It provides new operator norm estimates and weak bounds for these operators in weighted Lebesgue spaces, which were previously unknown.

Findings

Operator norms of the Hardy-Littlewood-Pólya operator derived

Weak bounds for multilinear n-dimensional integral operator established

Results applicable to weighted Lebesgue and weak Lebesgue spaces

Abstract

In this paper, we first obtain the operator norms of the $n$-dimensional Hardy-Littlewood-P\'{o}lya operator $\mathcal{H}$ from weighted Lebesgue spaces $L^p( \mathbb{R} ^n,| x |^{\beta} ) $ to weighted weak Lebesgue spaces $L^{q,\infty}(\mathbb{R} ^n,|x|^{\gamma})$. Next, we obtain the weak bounds for the $m$-linear $n$-dimensional integral operator.