Sharp bilinear estimates for maximal singular integrals with kernels in weighted $L^q$ spaces
Stefanos Lappas, Bae Jun Park
TL;DR
This work analyzes the dyadic maximal bilinear operator associated with rough kernels on $\mathbb{R}$ and proves sharp $L^{p_1}(\mathbb{R})\times L^{p_2}(\mathbb{R})\to L^{p}(\mathbb{R})$ bounds in the full quasi-Banach range, allowing $\Omega$ to lie in weighted $L^q$ spaces on $\mathbb{S}^1$. The authors decompose the maximal operator into a rough-part $\mathcal{E}^*$ and a smooth-part $\mathcal{L}_{\Omega}^{\sharp}$, develop dyadic kernel decompositions, and derive endpoint and interpolation estimates. By employing wavelet-based representations, a dyadic Duoandikoetxea–Rubio de Francia framework, and the Cao–Olivo–Yabuta weighted interpolation, they achieve the sharp condition $A>\frac{1}{p}+\frac{1}{q}-2$, unifying and extending prior 1D results of Honzík, Slavíková, Park and colleagues. The results advance understanding of bilinear maximal operators with rough kernels in weighted spaces and provide a robust set of techniques for handling quasi-Banach ranges and angular regularity.
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 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 $Ω$ of the kernel to belong to weighted $L^q$-spaces on $\mathbb{S}^1$.