Sparse domination for rough multilinear singular integrals

Abstract

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} $.