Weighted estimates for Multilinear Singular Integrals with Rough Kernels

Abstract

We establish weighted norm inequalities for multilinear singular integral operators with rough kernels. Specifically, we consider the multilinear singular integral operator ${\mathcal{L}}_{\hat{I}}\text{c◯}$ associated with an integrable function $\hat{I}\text{c◯}$ 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 ${\mathcal{L}}_{\hat{I}}\text{c◯}$ is bounded from $L^{p_1}(w_1)\times \cdots \times L^{p_m}(w_m)$ to $L^p(v_{\vec{\mathbf{w}}})$ under the assumption that $\hat{I}\text{c◯}\in L^q({\mathbb{S}}^{mn-1})$ and that the $m$-tuple of weights $\vec{\mathbf{w}}=(w_1,\dots ,w_m)$ lies in the multiple weight class ${\mathrm{A}}_{\vec{\mathbf{p}}/q^{\prime }}$. Here, $q^{\prime }$ denotes the Hölder conjugate of $q$, and we assume $q^{\prime }\le p_1,\dots ,p_m<\infty$ with $1/p=1/p_1+\cdots +1/p_m$.

Typos corrected and some details in the proofs added for clarity