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, He, and Slavíková (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 $L^{p_1}\times L^{p_2}\to L^p$ estimates for ${\mathcal{L}}_{\Omega }$ under the assumption $$\Omega \in L^q({\mathbb{S}}^{2n-1})\text{ for }\max (\frac{4}{3},\frac{p}{2p-1})<q\le \infty$$ 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 $\Omega \in L^{\infty }({\mathbb{S}}^{2n-1})$ is required for the $L^{p_1}\times L^{p_2}\to L^p$ boundedness.

Minor revision. To appear in Math. Ann