Weighted Norm Inequalities for Maximal Multi-sublinear Rough Singular Integrals

We continue our study of multilinear rough homogeneous singular integrals by establishing weighted estimates for their maximal truncations. Let $m\ge 2$ and let $\mathrm{\Omega }\in L^q({\mathbb{S}}^{mn-1})$, $1<q\le \mathrm{\infty }$, have mean zero. For $1<p_1,\dots ,p_m<\mathrm{\infty }$ with $p_j\ge q^{\mathrm{\prime }}$ and $1\mathrm{/}p={\sum }_{j=1}^{m}1\mathrm{/}{p}_j$, we prove $$\parallel {\mathcal{L}}_{\mathrm{\Omega }}^{\ast }(f_1,\dots ,f_m){\parallel }_{L^p(v_{\vec{\mathit{w}}})}\lesssim \mathrm{\parallel }\mathrm{\Omega }{\mathrm{\parallel }}_{L^q({\mathbb{S}}^{mn-1})}\prod _{j=1}^m\mathrm{\parallel }{f}_j{\mathrm{\parallel }}_{L^{p_j}(w_j)}$$ whenever $\vec{\mathit{w}}=(w_1,\dots ,w_m)\in A_{(p_1\mathrm{/}{q}^{\mathrm{\prime }},\dots ,p_m\mathrm{/}{q}^{\mathrm{\prime }})}$, where $q^{\mathrm{\prime }}$ is the Hölder conjugate of $q$ and $v_{\vec{\mathit{w}}}={\prod }_{j=1}^m{w}_j^{p\mathrm{/}{p}_j}$. The estimate holds under the same multiple weight condition as the corresponding principal value estimate.