  We obtain a sharp $L^2\times L^2 \to L^1$ boundedness criterion for a class
of bilinear operators associated with a multiplier given by a signed sum of
dyadic dilations of a given function, in terms of the $L^q$ integrability of
this function; precisely we show that boundedness holds if and only if $q<4$.
We discuss applications of this result concerning bilinear rough singular
integrals and bilinear dyadic spherical maximal functions.
  Our second result is an optimal $L^2\times L^2\to L^1$ boundedness criterion
for bilinear operators associated with multipliers with $L^\infty$ derivatives.
This result provides the main tool in the proof of the first theorem and is
also manifested in terms of the $L^q$ integrability of the multiplier. The
optimal range is $q<4$ which, in the absence of Plancherel's identity on $L^1$,
should be compared to $q=\infty$ in the classical $L^2\to L^2$ boundedness for
linear multiplier operators.
