Bilinear Rough Singular Integrals near the Critical Integrability via Sharp Fourier Multiplier Criteria

Abstract

We establish boundedness results for bilinear singular integral operators with rough homogeneous kernels whose restriction to the unit sphere belongs to the Orlicz space $L(\log L)^\alpha$. This improves the previously best known condition for boundedness of such bilinear operators obtained in the paper of the first and third authors, and provides estimates close to the conjectured endpoint of integrability suggested by the linear theory. The proof is based on a new sharp boundedness criterion for bilinear Fourier multiplier operators associated with sums of dyadic dilations of a fixed symbol $m_0$, compactly supported away from the origin. This criterion admits the best possible behavior with respect to a modulation of $m_0$ and is intimately connected with sharp shifted square function estimates.