Let $\Omega$ be a function on $\mathbb{ R}^{ mn} $, homogeneous of degree zero, and satisfy a cancellation condition on the unit sphere $\mathbb{ S}^{ mn-1}$. In this paper, we show that the multilinear singular integral operator \[ \mathcal{ T}_{ \Omega}(f_1, \ldots, f_m)(x) := \mathrm{ p.v.} \int_{ \mathbb{ R}^{ mn}} \frac{ \Omega(x - y_1, \ldots, x - y_m)}{ |x - \vec{ y}|^{ mn}} \prod_{ i=1}^m f_i(y_i) \, d\vec{ y}, \] associated with a rough kernel $\Omega \in L^r(\mathbb{ S}^{ mn-1}) $, $r > 1 $, admits a sparse domination, where $\quad \vec{ y}=(y_1,\ldots,y_m)$ and $ d\vec{ y}=dy_1\cdots dy_m$. As a consequence, we derive some { quantitative weighted norm inequalities} for $ \mathcal{ T}_{ \Omega} $.
Categories: math.CA