Limited smoothness conditions with mixed norms for bilinear Fourier multipliers

TL;DR

This paper investigates the boundedness of bilinear Fourier multiplier operators under weak smoothness conditions, extending classical results to more general multipliers with limited smoothness and vanishing conditions.

Contribution

It introduces new boundedness results for bilinear Fourier multipliers with limited smoothness, including applications to operators involving BMO and Hardy spaces.

Findings

Established $L^2 imes L^{ abla} o L^2$ boundedness under weak smoothness

Proved $L^2 imes L^2 o H^1$ boundedness with vanishing conditions

Extended classical multiplier theorems to multipliers with limited smoothness

Abstract

In this paper, the $L^2 \times L^{\infty} \to L^2$ and $L^2 \times L^2 \to L^1$ boundedness of bilinear Fourier multiplier operators is discussed under weak smoothness conditions on multipliers. As an application, we prove the $L^2 \times BMO \to L^2$ and $L^2 \times L^2 \to H^1$ boundedness of bilinear operators with multipliers of limited smoothness satisfying vanishing conditions.