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

TL;DR

This paper establishes sharp boundedness estimates for dyadic maximal bilinear operators with rough kernels in weighted $L^q$ spaces, extending previous results to a broader range of exponents and kernel conditions.

Contribution

It provides the first sharp $L^{p_1} imes L^{p_2} o L^{p}$ estimates for bilinear maximal operators with kernels in weighted $L^q$ spaces on the sphere.

Findings

Sharp $L^{p_1} imes L^{p_2} o L^{p}$ bounds established

Extended the range of exponents for boundedness in weighted $L^q$ spaces

Unified previous results in bilinear and one-dimensional settings

Abstract

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 $\mathbb{S}^1$.