Bilinear singular integral operators with kernels in weighted spaces

We establish the full quasi-Banach range of $L^{p_1}(\mathbb{R})\times L^{p_2}(\mathbb{R})\to L^p(\mathbb{R})$ bounds for one-dimensional bilinear singular integral operators with homogeneous kernels whose restriction $\mathrm{\Omega }$ to the unit sphere ${\mathbb{S}}^1$ is supported away from the degenerate line ${\theta }_1={\theta }_2$, 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 $\mathrm{\Omega }$ and requiring that $\mathrm{\Omega }\in L^q({\mathbb{S}}^1,u^q)$, where $u({\theta }_1,{\theta }_2)=\mathrm{\mid }{\theta }_1-{\theta }_2{\mathrm{\mid }}^{-1}$ for $({\theta }_1,{\theta }_2)\in {\mathbb{S}}^1$. In addition, we provide counterexamples that show the failure of the $n$-dimensional version of the previous result when $n\ge 2$, as well as the failure of its $m$-linear variant in dimension one when $m\ge 3$. The relationship of these results to (un)boundedness properties of higher-dimensional multilinear Hilbert transforms is also discussed.