The sharp weighted bound for multilinear maximal functions and Calder\'{o}n-Zygmund operators

We investigate the weighted bounds for multilinear maximal functions and Calder'on-Zygmund operators from $L^{p_1}(w_1)\times \mathrm{.}\mathrm{.}\mathrm{.}\times L^{p_m}(w_m)$ to $L^p(v_{\vec{w}})$, where $1<p_1,\mathrm{.}\mathrm{.}\mathrm{.},p_m<\mathrm{\infty }$ with $1\mathrm{/}{p}_1+\mathrm{.}\mathrm{.}\mathrm{.}+1\mathrm{/}{p}_m=1\mathrm{/}p$ and $\vec{w}$ is a multiple $A_{\vec{P}}$ weight. We prove the sharp bound for the multilinear maximal function for all such $p_1,\mathrm{.}\mathrm{.}\mathrm{.},p_m$ and prove the sharp bound for $m$-linear Calder'on-Zymund operators when $p\ge 1$.