Fourier Multipliers on the Heisenberg groups revisited

TL;DR

This paper provides explicit formulas and a simplified proof for Fourier multiplier theorems on Heisenberg groups, along with sharp weighted estimates, advancing the understanding of harmonic analysis on these groups.

Contribution

It offers explicit differential-difference operators, a shorter proof of the Fourier multiplier theorem, and sharp weighted estimates for Heisenberg groups.

Findings

Explicit expressions for differential-difference operators

A shorter proof of the Fourier multiplier theorem

Sharp weighted estimates for Fourier multipliers

Abstract

In this paper, we give explicit expressions of differential-difference operators appeared in the hypothesis of the general Fourier multiplier theorem associated to the Heisenberg groups proved by Mauceri and De Micheal for one dimension and C. Lin for higher dimension. We also give a much shorter proof of the above-mentioned theorem. Then we obtain a sharp weighted estimate for Fourier multipliers on the Heisenberg groups.