Strongly singular bilinear Calder\'on-Zygmund operators and a class of bilinear pseudodifferential operators

TL;DR

This paper studies the boundedness of strongly singular bilinear Calderón-Zygmund operators, establishing their properties in Lebesgue, BMO, and Hardy spaces, and providing weighted boundedness results under kernel conditions.

Contribution

It introduces new boundedness results for strongly singular bilinear Calderón-Zygmund operators, including endpoint and weighted estimates, based on integral and pointwise kernel conditions.

Findings

Boundedness in Lebesgue spaces

Endpoint mappings involving BMO and Hardy spaces

Weighted Lebesgue space boundedness

Abstract

Motivated by the study of kernels of bilinear pseudodifferential operators with symbols in a H\"ormander class of critical order, we investigate boundedness properties of strongly singular Calder\'on--Zygmund operators in the bilinear setting. For such operators, whose kernels satisfy integral-type conditions, we establish boundedness properties in the setting of Lebesgue spaces as well as endpoint mappings involving the space of functions of bounded mean oscillations and the Hardy space. Assuming pointwise-type conditions on the kernels, we show that strongly singular bilinear Calder\'on--Zygmund operators satisfy pointwise estimates in terms of maximal operators, which imply their boundedness in weighted Lebesgue spaces.