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

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