An alternative to Plancherel’s criterion for bilinear operators

We prove that bilinear operators associated with $L^q$ multipliers with sufficiently many derivatives in $L^\infty $ are bounded from $L^2\times L^2$ to $L^1$ when $q \lt 4$. In the absence of Plancherel’s identity on $L^1$, the range $q \lt 4$ in the bilinear case should be compared to $q=\infty $ in the classical $L^2\to L^2$ boundedness for linear multiplier operators.