Sparse domination for rough multilinear singular integrals

Abstract

Let $\hat{I}\text{c◯}$ 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 \[ \mathcal{T}_Ω(f_1, \ldots, f_m)(x) := \mathrm{p.v.} \int_{\mathbb{R}^{mn}} \frac{Ω(x - y_1, \ldots, x - y_m)}{|x - \vec{y}|^{mn}} \prod_{i=1}^m f_i(y_i) \, d\vec{y}, \] associated with a rough kernel $\hat{I}\text{c◯}\in L^r({\mathbb{S}}^{mn-1})$, $r>1$, admits a sparse domination, where $\vec{y}=(y_1,\dots ,y_m)$ and $d\vec{y}=d{y}_1\cdots d{y}_m$. As a consequence, we derive some {quantitative weighted norm inequalities} for ${\mathcal{T}}_{\hat{I}}\text{c◯}$.

13 pages