Sharp weak bounds for the Hardy-Littlewood-Pólya operator and the weak bounds for the multilinear integral operator

Abstract

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