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

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.