Weighted Norm Inequalities for Maximal Multi-sublinear Rough Singular Integrals

Abstract

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