Sparse domination and weighted estimates for rough bilinear singular integrals

Abstract

Let $r>\frac{4}{3}$ and let $\Omega \in L^r({\mathbb{S}}^{2n-1})$ have vanishing integral. We show that the bilinear rough singular integral $$T_{\Omega }(f,g)(x)=\text{p.v.}{\int }_{{\mathbb{R}}^n}{\int }_{{\mathbb{R}}^n}\frac{\Omega ((y,z)/|(y,z)|)}{|(y,z){|}^{2n}}f(x-y)g(x-z)dydz,$$ satisfies a sparse bound by $(p,p,p)$-averages, where $p$ is bigger than a certain number explicitly related to $r$ and $n$. As a consequence we deduce certain quantitative weighted estimates for bilinear homogeneous singular integrals associated with rough homogeneous kernels.

arXiv comment22 pages

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