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 $m\ge2$ and let $Ω\in L^q(\mathbb{S}^{mn-1})$, $1<q\le\infty$, have mean zero. For $1<p_1,\dots,p_m<\infty$ with $p_j\ge q&#39;$ and $1/p=\sum_{j=1}^m1/p_j$, we prove $$\big\Vert \mathcal{L}_Ω^*(f_1,\dots,f_m)\big\Vert_{L^p(v_{\vec{\boldsymbol{w}}})} \lesssim \VertΩ\Vert_{L^q(\mathbb{S}^{mn-1})} \prod_{j=1}^m\Vert f_j\Vert_{L^{p_j}(w_j)}$$ whenever $\vec{\boldsymbol{w}}=(w_1,\dots,w_m)\in A_{(p_1/q&#39;,\dots,p_m/q&#39;)}$, where $q&#39;$ is the Hölder conjugate of $q$ and $v_{\vec{\boldsymbol{w}}}=\prod_{j=1}^m w_j^{p/p_j}$. The estimate holds under the same multiple weight condition as the corresponding principal value estimate.