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Improved estimates for bilinear rough singular integrals

We study bilinear rough singular integral operators $\mathcal{L}_Ω$ associated with a function $Ω$ on the sphere $\mathbb{S}^{2n-1}$. In the recent work of Grafakos, He, and Slavíková (Math. Ann. 376: 431-455, 2020), they showed that $\mathcal{L}_Ω$ is bounded from $L^2\times L^2$ to $L^1$, provided that $Ω\in L^q(\mathbb{S}^{2n-1})$ for $4/3<q\le \infty$ with mean value zero. In this paper, we provide a generalization of their result. We actually prove $L^{p_1}\times L^{p_2}\to L^p$ estimates for $\mathcal{L}_Ω$ under the assumption $$Ω\in L^q(\mathbb{S}^{2n-1}) \quad \text{ for }~\max{\Big(\;\frac{4}{3}\;,\; \frac{p}{2p-1} \;\Big)<q\le \infty}$$ where $1<p_1,p_2\le\infty$ and $1/2<p<\infty$ with $1/p=1/p_1+1/p_2$ . Our result improves that of Grafakos, He, and Honzík (Adv. Math. 326: 54-78, 2018), in which the more restrictive condition $Ω\in L^{\infty}(\mathbb{S}^{2n-1})$ is required for the $L^{p_1}\times L^{p_2}\to L^p$ boundedness.