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.

我们研究带有符号的所有（或足够多）阶有界偏导数的双线性傅立叶乘法算子的两种有界条件。第一种定理明确规定了符号本身的某种衰减率，而第二种定理还要求符号的所有导数具有相同的衰减率。我们表明，即使这两种类型的双线性乘法定理密切相关，但在极限情况下它们之间还是存在一些根本性的区别。另外，由于到目前为止，主要是针对更一般的双线性伪微分算子类型研究了后一种类型的定理，因此在双线性傅立叶乘法器的特殊情况下，我们将重新讨论它们，