The H\"ormander multiplier theorem, II: The bilinear local $L^2$ case

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 $n\mathrm{/}2$ derivatives. We also provide counterexamples to obtain necessary conditions for all sets of indices.