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 $1 < p_1, p_2 < \infty$ and $1/2 < p < \infty$. Our approach extends and unifies several recent contributions, including those of Honz\'ik, the first author, and Slav\'ikova, as well as the second author in the bilinear and in the one-dimensional settings, by allowing the angular component $\Omega$ of the kernel to belong to weighted $L^q$-spaces on $\mathbb{S}^1$.