  We use wavelets of tensor product type to obtain the boundedness of bilinear
multiplier operators on $\mathbb R^n\times \mathbb R^n$ associated with
H\"ormander multipliers on $\mathbb R^{2n}$ with minimal smoothness. We focus
on the local $L^2$ case and we obtain boundedness under the minimal smoothness
assumption of $n/2$ derivatives. We also provide counterexamples to obtain
necessary conditions for all sets of indices.
