Bilinear singular integral operators with kernels in weighted spaces

Abstract

We establish the full quasi-Banach range of $L^{p_1}(\mathbb R) \times L^{p_2}(\mathbb R) \rightarrow L^p(\mathbb R)$ bounds for one-dimensional bilinear singular integral operators with homogeneous kernels whose restriction $Ω$ to the unit sphere $\mathbb S^1$ is supported away from the degenerate line $θ_1=θ_2$, belongs to $L^q(\mathbb S^1)$ for some $q>1$ and has vanishing integral. In fact, a more general result is obtained by dropping the support condition on $Ω$ and requiring that $Ω\in L^q(\mathbb S^1,u^q)$, where $u(θ_1,θ_2)=|θ_1-θ_2|^{-1}$ for $(θ_1,θ_2)\in \mathbb S^1$. In addition, we provide counterexamples that show the failure of the $n$-dimensional version of the previous result when $n\geq 2$, as well as the failure of its $m$-linear variant in dimension one when $m\geq 3$. The relationship of these results to (un)boundedness properties of higher-dimensional multilinear Hilbert transforms is also discussed.