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 $$\begin{aligned} T_{\Omega }(f,g)(x)= \text {p.v.} \int _{\mathbb {R}^n}\int _{\mathbb {R}^n}\frac{\Omega ((y,z)/|(y,z)|)}{|(y,z)|^{2n}}f(x-y)g(x-z)\,dydz, \end{aligned}$$ 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.