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 $\mathcal{L}_\Omega$ associated with an integrable function $\Omega$ 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 $\mathcal{L}_\Omega$ is bounded from $L^{p_1}(w_1)\times\cdots\times L^{p_m}(w_m)$ to $L^p(v_{\vec{\boldsymbol{w}}})$ under the assumption that $\Omega\in L^q(\mathbb{S}^{mn-1})$ and that the $m$ tuple of weights $\vec{\boldsymbol{w}}= (w_1,\ldots,w_m)$ lies in the multiple weight class $A_{\vec{\boldsymbol{p}}/q'}((\mathbb{R}^n)^m)$. Here, $q'$ denotes the H\"older conjugate of $q$, and we assume $q'\le p_1,\dots,p_m<\infty$ with $1/p = 1/p_1 + \cdots + 1/p_m$.