Improved estimates for bilinear rough singular integrals

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