  We investigate two types of boundedness criteria for bilinear Fourier
multiplier operators with symbols with bounded partial derivatives of all (or
sufficiently many) orders. Theorems of the first type explicitly prescribe only
a certain rate of decay of the symbol itself while theorems of the second type
require, in addition, the same rate of decay of all derivatives of the symbol.
We show that even though these two types of bilinear multiplier theorems are
closely related, there are some fundamental differences between them which
arise in limiting cases. Also, since theorems of the latter type have so far
been studied mainly in connection with the more general class of bilinear
pseudodifferential operators, we revisit them in the special case of bilinear
Fourier multipliers, providing also some improvements of the existing results
in this setting.
