Sparse domination for rough multilinear singular integrals

TL;DR

This paper proves that rough multilinear singular integrals can be dominated by sparse operators, leading to new weighted norm inequalities, advancing understanding of such operators with minimal smoothness assumptions.

Contribution

It establishes sparse domination for rough multilinear singular integrals with kernels in L^r, enabling new weighted inequalities and extending prior smooth kernel results.

Findings

Sparse domination holds for rough kernels in L^r

Derived quantitative weighted norm inequalities

Extended the theory to less smooth kernels

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} $.