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We obtain a sharp $L^2\times L^2 \to L^1$ boundedness criterion for a classof bilinear operators associated with a multiplier given by a signed sum ofdyadic dilations of a given function, in terms of the $L^q$ integrability ofthis function; precisely we show that boundedness holds if and only if $q<4$.We discuss applications of this result concerning bilinear rough singularintegrals and bilinear dyadic spherical maximal functions.  Our second result is an optimal $L^2\times L^2\to L^1$ boundedness criterionfor bilinear operators associated with multipliers with $L^\infty$ derivatives.This result provides the main tool in the proof of the first theorem and isalso manifested in terms of the $L^q$ integrability of the multiplier. Theoptimal 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 forlinear multiplier operators.

$L^2\times L^2 \to L^1$ boundedness criteria