Homogeneous Fourier and Weyl multipliers on Sobolev spaces related to the Heisenberg group

TL;DR

This paper investigates the limitations of homogeneous Fourier and Weyl multipliers on Sobolev spaces associated with the Heisenberg group, extending prior results to a non-commutative setting.

Contribution

It establishes new results showing that non-constant homogeneous multipliers cannot act as Fourier or Weyl multipliers on these Sobolev spaces, generalizing classical theorems.

Findings

Homogeneous Fourier multipliers must be constant on the Heisenberg group.

Weyl multipliers with homogeneity degree zero are also necessarily constant.

Results extend classical Fourier multiplier theorems to non-commutative groups.

Abstract

Inspired by the work of A. Bonami and S. Poornima that a non-constant function which is homogeneous of degree $0$ cannot be a Fourier multiplier on homogeneous Sobolev spaces, we establish analogous results for Fourier multipliers on the Heisenberg group $ \mathbb{H}^n $ and Weyl multipliers on $ \mathbb{C}^n $ acting on Sobolev Spaces.