Multilinear estimates for maximal rough singular integrals

In this work, we establish $L^{p_1}\times \cdots \times L^{p_m}\to L^p$ bounds for maximal multi-(sub)linear singular integrals associated with homogeneous kernels $\frac{\mathrm{\Omega }({\vec{\mathit{y}}}^{\mathrm{\prime }})}{\mathrm{\mid }\vec{\mathit{y}}{\mathrm{\mid }}^{mn}}$ where $\mathrm{\Omega }$ is an $L^q$ function on the unit sphere ${\mathbb{S}}^{mn-1}$ with vanishing moment condition and $q>1$. As an application, we obtain almost everywhere convergence results for the associated doubly truncated multilinear singular integrals.