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 $\Omega$ 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 $\Omega$ and requiring that $\Omega\in L^q(\mathbb S^1,u^q)$, where $u(\theta_1,\theta_2)=|\theta_1-\theta_2|^{-1}$ 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 $n\geq 2$, 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.