Localization Operator and Weyl Transform on Reduced Heisenberg Group with Multi-dimensional Center

TL;DR

This paper investigates localization operators and Weyl transforms on a reduced Heisenberg group with a multidimensional center, establishing their properties and operator bounds based on symbol classes.

Contribution

It introduces definitions of localization and Weyl operators on the reduced Heisenberg group and analyzes their boundedness and compactness properties.

Findings

Weyl transform is bounded and compact for certain symbol classes

Weyl transform becomes unbounded when the symbol class parameter exceeds 2

Product formula for localization operators on the group

Abstract

In this article, we study two different types of operators, the localization operator and Weyl transform, on the reduced Heisenberg group with multidimensional center $\mathcal{G}$. The group $\mathcal{G}$ is a quotient group of non-isotropic Heisenberg group with multidimensional center $\mathcal{H}^m$ by its center subgroup. Firstly, we define the localization operator using a wavelet transform on $\mathcal{G}$ and obtain the product formula for the localization operators. Next, we define the Weyl transform associated to the Wigner transform on $\mathcal{G}$ with the operator-valued symbol. Finally, we have shown that the Weyl transform is not only a bounded operator but also a compact operator when the operator-valued symbol is in $L^p,1\leq p\leq 2,$ and it is an unbounded operator when $p>2$.