  This article is concerned with the question of whether Marcinkiewicz
multipliers on $\mathbb R^{2n}$ give rise to bilinear multipliers on $\mathbb
R^n\times \mathbb R^n$. We show that this is not always the case. Moreover we
find necessary and sufficient conditions for such bilinear multipliers to be
bounded. These conditions in particular imply that a slight logarithmic
modification of the Marcinkiewicz condition gives multipliers for which the
corresponding bilinear operators are bounded on products of Lebesgue and Hardy
spaces.
