  Boundedness properties for pseudodifferential operators with symbols in the
bilinear H\"ormander classes of sufficiently negative order are proved. The
results are obtained in the scale of Lebesgue spaces and, in some cases,
end-point estimates involving weak-type spaces and BMO are provided as well.
From the Lebesgue space estimates, Sobolev ones are then easily obtained using
functional calculus and interpolation. In addition, it is shown that, in
contrast with the linear case, operators associated with symbols of order zero
may fail to be bounded on products of Lebesgue spaces.
