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 $\frac{\Omega(\vec{\boldsymbol{y}}')}{|\vec{\boldsymbol{y}}|^{mn}}$ where $\Omega$ 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.