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

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.