Weighted estimates for Multilinear Singular Integrals with Rough Kernels

We establish weighted norm inequalities for multilinear singular integral operators with rough kernels. Specifically, we consider the multilinear singular integral operator ${\mathcal{L}}_{\mathrm{\Omega }}$ associated with an integrable function $\mathrm{\Omega }$ on the unit sphere ${\mathbb{S}}^{mn-1}$ satisfying the vanishing mean condition. Extending the classical results of Watson and Duoandikoetxea to the multilinear setting, we prove that ${\mathcal{L}}_{\mathrm{\Omega }}$ is bounded from $L^{p_1}(w_1)\times \cdots \times L^{p_m}(w_m)$ to $L^p(v_{\vec{\mathit{w}}})$ under the assumption that $\mathrm{\Omega }\in L^q({\mathbb{S}}^{mn-1})$ and that the $m$ tuple of weights $\vec{\mathit{w}}=(w_1,\dots ,w_m)$ lies in the multiple weight class $A_{\vec{\mathit{p}}\mathrm{/}{q}^{\mathrm{\prime }}}(({\mathbb{R}}^n{)}^m)$. Here, $q^{\mathrm{\prime }}$ denotes the Hölder conjugate of $q$, and we assume $q^{\mathrm{\prime }}\le p_1,\dots ,p_m<\mathrm{\infty }$ with $1\mathrm{/}p=1\mathrm{/}{p}_1+\cdots +1\mathrm{/}{p}_m$.