  Results analogous to those proved by Rubio de Francia are obtained for a
class of maximal functions formed by dilations of bilinear multiplier operators
of limited decay. We focus our attention to $L^2\times L^2\to L^1$ estimates.
We discuss two applications: the boundedness of the bilinear maximal
Bochner-Riesz operator and of the bilinear spherical maximal operator. For the
latter we improve the known results by reducing the dimension restriction from
$n\ge 8$ to $n\ge 4$.
