Quaternion Weyl Transform and some uniqueness results

TL;DR

This paper investigates the properties of the quaternion Weyl transform, establishing its compactness for certain symbols, its limitations in boundedness, and presenting a uniqueness theorem related to its rank, with implications for quaternion analysis.

Contribution

It introduces the quaternion Weyl transform, proves its compactness for specific symbol classes, and establishes a rank-based uniqueness theorem, extending classical results to quaternionic settings.

Findings

QWT is compact for symbols in $L^{r}$ with $1 \,\leq r \leq 2$.

QWT cannot be extended as a bounded operator for $r>2$.

A rank analogue of the Benedicks-Amrein-Berthier theorem is proved for QWT.

Abstract

In this article, we study the boundedness and several properties of the quaternion Wigner transform. Using the quaternion Wigner transform as a tool, we define the quaternion Weyl transform (QWT) and prove that the QWT is compact for a certain class of symbols in $L^{r}\left(\mathbb{R}^{4}, \mathbb{Q}\right)$ with $1 \leq r \leq 2.$ Moreover, it can not be extended as a bounded operator for symbols in $L^{r}\left(\mathbb{R}^{4},\mathbb{Q}\right)$ for $2<r<\infty.$ In addition, we prove a rank analogue of the Benedicks-Amrein-Berthier theorem for the QWT. Further, we remark about the set of injectivity and Helgason's support theorem for the quaternion twisted spherical means.