The Hörmander 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 Hormander 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.