Bilinear singular integral operators with kernels in weighted spaces

TL;DR

This paper establishes boundedness results for one-dimensional bilinear singular integral operators with kernels in weighted spaces, extending previous results and providing counterexamples in higher dimensions and multilinear cases.

Contribution

It extends the boundedness range for bilinear singular integrals with kernels in weighted spaces and introduces counterexamples for higher-dimensional and multilinear variants.

Findings

Full quasi-Banach range of bounds established for 1D bilinear operators.

Counterexamples show failure of similar bounds in higher dimensions and for multilinear cases.

Relationship to higher-dimensional multilinear Hilbert transforms discussed.

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.