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 $L^p({\mathbb{R}}^n,|x{|}^{\beta })$ 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.

There is a critical derivation error in Section 3 that affects the main conclusion. I withdraw this version to correct the error and will submit a revised version later