Improved estimates for bilinear rough singular integrals

Abstract

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