Bilinear Fourier multipliers and the rate of decay of their derivatives

TL;DR

This paper examines boundedness criteria for bilinear Fourier multipliers, focusing on how the decay rate of symbols and their derivatives affects operator boundedness, and clarifies differences between two main theorem types.

Contribution

It compares two classes of bilinear multiplier theorems based on decay conditions, highlighting fundamental differences and improving results for Fourier multipliers.

Findings

Theorem types differ in their decay requirements for symbols and derivatives.

Fundamental differences arise in limiting cases between the two theorem types.

Improved results are provided for bilinear Fourier multipliers in this study.

Abstract

We investigate two types of boundedness criteria for bilinear Fourier multiplier operators with symbols with bounded partial derivatives of all (or sufficiently many) orders. Theorems of the first type explicitly prescribe only a certain rate of decay of the symbol itself while theorems of the second type require, in addition, the same rate of decay of all derivatives of the symbol. We show that even though these two types of bilinear multiplier theorems are closely related, there are some fundamental differences between them which arise in limiting cases. Also, since theorems of the latter type have so far been studied mainly in connection with the more general class of bilinear pseudodifferential operators, we revisit them in the special case of bilinear Fourier multipliers, providing also some improvements of the existing results in this setting.