  The reappearance of a sometimes called exotic behavior for linear and
multilinear pseudodifferential operators is investigated. The phenomenon is
shown to be present in a recently introduced class of bilinear
pseudodifferential operators which can be seen as more general variable
coefficient counterparts of the bilinear Hilbert transform and other singular
bilinear multipliers operators. The unboundedness on product of Lebesgue spaces
but the boundedness on spaces of smooth functions (which is the exotic behavior
referred to) of such operators is obtained. In addition, by introducing a new
way to approximate the product of two functions, estimates on a new
paramultiplication are obtained.
