Sharp bilinear estimates for maximal singular integrals with kernels in weighted $L^q$ spaces

In this paper, we study the boundedness properties of the (dyadic) maximal bilinear operator associated with rough homogeneous kernels on $\mathbb{R}$. We establish sharp $L^{p_1}(\mathbb{R})\times L^{p_2}(\mathbb{R})\to L^p(\mathbb{R})$ estimates in the full quasi-Banach range of exponents $1<p_1,p_2<\mathrm{\infty }$ and $1\mathrm{/}2<p<\mathrm{\infty }$. Our approach extends and unifies several recent contributions, including those of Honzík, the first author, and Slavíkova, as well as the second author in the bilinear and in the one-dimensional settings, by allowing the angular component $\mathrm{\Omega }$ of the kernel to belong to weighted $L^q$-spaces on ${\mathbb{S}}^1$.