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

TL;DR

This paper establishes boundedness of bilinear multiplier operators on  spaces using wavelets, requiring minimal smoothness, and provides counterexamples for necessary conditions on indices.

Contribution

It extends the Hf6rmander multiplier theorem to the bilinear local L^2 case with minimal smoothness assumptions and counterexamples for index conditions.

Findings

Boundedness of bilinear multipliers with n/2 derivatives smoothness.

Use of tensor product wavelets for analysis.

Counterexamples for index necessity.

Abstract

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.